{"version":1,"lectureId":"01M1T40VFP796V47RK8BPV01XA","attempt":0,"publication":{"slug":"understanding-graphical-projections-perspective-oblique-and-orthographic","title":"Understanding Graphical Projections: Perspective, Oblique, and Orthographic","subject":"engineering","summary":"Every drawing of a solid object begins with the same decision: which straight rays carry the object onto the flat page. This lecture follows that decision through the three systems that matter. Perspective sends every ray to a single eye, so distant things shrink and parallel rails meet at a vanishing point: the drawing looks right and cannot be measured. Oblique and orthographic projection use parallel rays instead, trading optical realism for true lengths, true angles, and one scale a ruler can use. We derive the inverse distance rule of perspective, build cavalier and cabinet oblique from a single slant angle, collapse a solid into three orthographic views, reach the isometric view by turning the object, and finish by showing that a parallel projection is simply a perspective seen from infinitely far away.","metaDescription":"How perspective, oblique, orthographic and isometric projections work, why perspective looks real, and which drawings you can measure.","transcript":"Here is a solid block. It has width, it has depth, and it has height: three directions at once, and six flat faces. And here is the difficulty. A page is flat. It has two directions to spend, and the block has three, so one of those three cannot be drawn as it truly is. Every drawing system in this lecture is a different answer to that. So look down on the whole business from above. The page stands edge on, as this pale vertical line. The object sits over here, and the drawing will appear where the rays cross it. Suppose every ray runs to one single point, an eye out at the front. From each corner of the object, a straight ray to that point, and a mark where it crosses. Between the two marks is the drawing, and its height is h prime. Now compare the two heights. The object stands this tall, and its drawing on the page is well under half of that, because rays that converge shrink whatever is far away. That shrinking is perspective, and it is exactly what your own eye does. Now change one thing only. Take the eye away, and let every ray run parallel to its neighbours instead of gathering to a point. Nothing converges now, so nothing shrinks. The drawing grows until it is exactly as tall as the object itself, and a ruler laid on the page measures the real thing. So there are two great families of projection, and every drawing ever made belongs to one of them. Rays that meet at a point, or rays that never meet at all. The converging family has just one member, and it is called perspective. The parallel family splits in two, by the angle at which the rays strike the page. At a slant, the drawing is called oblique. Square on, it is orthographic, and that includes the axonometric views, of which isometric is the one you have certainly seen. Perspective, oblique, orthographic. One of the three shows the world as an eye receives it. The other two show something no eye can ever see, and that is exactly why they are what you build from. Start with the converging family. One eye, one page, one post. The eye is here, and the page stands at a fixed distance in front of it, which I will call f. The post stands out at a distance Z, and it is h tall. Now the two rays. One from its top to the eye, one from its foot to the eye. Each crosses the page on the way, and between the two crossings is the drawn image, of height h prime. Now the geometry does the work. The big triangle from the eye out to the post, and the small one from the eye to the page, share an apex and a direction, so they are similar. Similar triangles give a proportion: h prime over h equals f over Z. Rearrange that, and the drawn height is f times h over Z. There is the whole of perspective in a single line: what you draw is inversely proportional to how far away the thing is. Put numbers on that. At one page distance out, the drawing is life size. At twice the distance, half size. At four times out, a quarter. And here it is happening. The post walks out, and its mark on the page closes down. Further still, and smaller again. Bring it back in, and the mark swells. Now the famous consequence. Take two rails that really are parallel, and really are the same distance apart at every step of the way. Their images do not stay apart. Every ray to the eye pulls them closer, and on the page they meet, at a single point on the horizon. That is the vanishing point, and nothing whatever in the world stands there. Now lay sleepers across the track, equally spaced, one every metre. Out in the world every gap is the same. On the page they crowd together as they go back, and the last of them are almost on top of each other. So a perspective drawing is faithful to the eye and useless to a ruler. Two marks of the same length on the page can be a metre apart in the world, or a hundred, and the drawing will not tell you which. Which is why perspective belongs to painting, to photography, to film, and to the camera in every video game: all the places where the job is to look right. It is not where you go when the job is to be built. Cross into the parallel family. Here is the page again, edge on, and the object behind it, and this time the rays arrive at a slant. Set the front face of the object flat against the page. Those points are already on it, so they draw themselves: same size, same shape, same angles, exactly. Now follow the rear face. Its rays are slanted, so they land further along the page, and every one of its points is shifted by the very same amount, because the rays never converge. That shift is the depth of the drawing, and the slant angle alone decides how big it is. Near and far are drawn at one single scale. Here is what that gives you on a cube. The front face is a true square: right angles, equal sides. Measure it on the page and you have measured the object. The depth runs off at forty five degrees, the same length from every corner, and the far face is a second true square, parallel to the first. Drawn at full depth, this is cavalier oblique, and every single edge on the page is a true length. But look at it: it seems to swell as it goes back, because your eye expects far things to be smaller, and this drawing refuses. So draughtsmen halve the depth. That is cabinet oblique: the front face still exactly true, the receding edges at half scale, and the solid suddenly looks like a solid. The price is easy to state. Only the face lying in the page can be measured directly, and only lines parallel to that face keep their length. Everything in the depth direction is a convention rather than a measurement. It is cheap to draw by hand and it reads at a glance, which is why oblique filled workshop notebooks and furniture drawings for two hundred years. The cabinet convention is named after cabinets. One change is left to make. Here are the slanted rays again, with the front face true on the page, and the rear face shifted along it. Now swing the rays square to the page, and watch the shift close up. There it goes: the rear face lands exactly on top of the front face, and the depth of the object has gone out of the drawing altogether. This is orthographic projection, and losing the depth is not a defect. What is left on the page is one face, dead true: true lengths, true angles, and every one of them measurable. But one view like this cannot describe a solid. A thin plate and a long block give exactly the same front view, so the answer is to draw more than one of them. So here is the standard set of three. The front view, straight on. Above it the top view, looking down from over the block. Beside it the side view, looking in from the right. And the three are locked together. Width is shared by the front and the top, so those two line up along here. Height is shared by the front and the side, so they line up across here. Between them, three flat drawings hold every dimension of the block, each one at true size. Nothing here looks like the object, and that is the whole trade. The drawing has stopped being a picture and become a measurement. This is the language that workshop drawings and building drawings are written in. There is a middle way, and you have seen it a thousand times. Keep the rays parallel and square to the page, exactly as they are now, but turn the block itself, so that three faces show at once. Now it reads as a solid again. But look along its edges: the ones that are parallel on the block are still parallel on the page. Nothing converges, so this is still a parallel projection, and it can still be measured. Choose the tilt so that the three axes are treated alike, and you have the isometric view. On the page those axes stand a hundred and twenty degrees apart, and all three are shortened by the very same factor, about zero point eight one six. That is the pair of properties that made it famous. It looks like a solid, and one ruler measures all three directions at once. Which is why isometric is the language of assembly instructions, of pipe runs, of patent drawings, and of a whole generation of video games. So which of these is real? There is one honest answer, and then a twist. Perspective is what an eye receives, because an eye has a single pupil, and every ray that reaches it converges there. A photograph is a perspective projection whether the photographer wanted one or not. Now the twist. Carry the eye backwards, and leave the object and the page exactly where they are. The rays flatten out. The image on the page grows, and the near and far parts of the object are treated more and more alike. Take that to its end. Put the eye infinitely far away, and the rays are exactly parallel. The shrinking stops altogether, and the mark on the page is precisely the size of the object. So a parallel projection is not a mistake about vision. It is the view from an infinite distance, which no eye can ever occupy. That is why an oblique or an isometric drawing always looks very slightly impossible. It is impossible on purpose. Then the useful question is not which one is real. It is what each one throws away, and who can afford to lose it. Perspective keeps the look of the world and throws measurement away. For a painting, a photograph, a film, or the camera in a game engine, that is exactly the right bargain. Oblique keeps one true face and throws the rest away. It is the quickest useful pictorial drawing there is, which is why it filled notebooks and furniture drawings for two hundred years. Orthographic throws the picture away entirely and keeps every size. Nothing is ever built from a perspective drawing. Buildings and machine parts are built from orthographic views. And isometric sits between the two. One scale for all three axes, a solid that reads as a solid, and a drawing you can still measure: assembly guides, pipe runs, and a great deal of game art. So there is the whole map, and one question decides it. What is the drawing for? If it must look right, let the rays converge. If it must be measured, keep them parallel, and then choose what stays true: one face for oblique, one view at a time for orthographic, all three axes alike for isometric.","watch":{"version":1,"scenes":[{"title":"From a Solid to a Flat Page","start":0,"end":144.40677083333333,"objects":{"block":"a Solid [blue] drawn in space (upper=<function>, x_range=(0.3, 2.3), y_range=(0.3, 1.9))","body":"a Polygon [blue] drawn in view (vertices=((4.0, 1.2), (5.0, 1.2), (5.0, 2.8), (4.0, 2.8)), fill_opacity=0.25)","body_tag":"a Math [blue] that says \"$upright(\"the object\")$\" drawn in view","eye":"a Point [red] labelled \"E\" drawn in view (location=(0.55, 2.0))","flat_hi":"an Arrow [yellow] drawn in view (start=(6.0, 2.8), end=(2.0, 2.8))","flat_lo":"an Arrow [yellow] drawn in view (start=(6.0, 1.2), end=(2.0, 1.2))","flat_mid":"an Arrow [yellow] drawn in view (start=(6.0, 2.0), end=(2.0, 2.0))","heading_block":"a Heading that says \"A Solid Body and a Flat Page\"","heading_rays":"a Heading that says \"Choose the Rays, Choose the Drawing\"","heading_three":"a Heading that says \"Three Families of Projection\"","image":"a Line [green] drawn in view (start=(2.0, (2.0 - reach)), end=(2.0, (2.0 + reach)))","image_tag":"a Math [green] that says \"$h'$\" drawn in view","item_obl":"a Text [text] that says \"Oblique: parallel rays, meeting the page at a slant.\"","item_orth":"a Text [text] that says \"Orthographic and axonometric: parallel rays, square on to the page.\"","item_persp":"a Text [text] that says \"Perspective: the rays all run to one eye.\"","page":"a Line [gray] drawn in view (start=(2.0, 0.5), end=(2.0, 3.4))","page_tag":"a Math [gray] that says \"$upright(\"the page\")$\" drawn in view","ray_hi":"a Line [yellow] drawn in view (start=(4.0, 2.8), end=(0.55, 2.0))","ray_lo":"a Line [yellow] drawn in view (start=(4.0, 1.2), end=(0.55, 2.0))","reach":"a VariableNumber (initial_value=0.336)","space":"an Axes3D (x_range=(0.0, 2.6), y_range=(0.0, 2.2), z_range=(0.0, 1.8))","tall":"a Math [blue] that says \"$h$\" drawn in view","verdict":"a Math [text] that says \"$h' < h$\"","view":"a Figure (x_range=(0.0, 6.1), y_range=(0.2, 3.9), aspect=(6.1, 3.7))"},"beats":[{"start":0,"say":"Here is a solid block. 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The object sits over here, and the drawing will appear where the rays cross it.","live":[],"does":[[25.380499999999998,"heading_rays is shown on the screen, written out."],[25.380499999999998,"view is shown on the screen, written out."],[31.950999999999997,"page is shown on the screen, written out."],[31.950999999999997,"page_tag is shown on the screen, written out."],[33.727999999999994,"body is shown on the screen, written out."],[33.727999999999994,"body_tag is shown on the screen, written out."]]},{"start":39.076499999999996,"say":"Suppose every ray runs to one single point, an eye out at the front. From each corner of the object, a straight ray to that point, and a mark where it crosses. Between the two marks is the drawing, and its height is h prime.","live":["view","heading_rays","page","page_tag","body","body_tag"],"does":[[41.143,"eye is shown on the screen, written out."],[45.206,"ray_hi is shown on the screen, written out."],[45.41363960391702,"ray_lo is shown on the screen, written out."],[52.149,"image is shown on the screen, written out."],[54.785000000000004,"image_tag is shown on the screen, written out."]]},{"start":56.2205,"say":"Now compare the two heights. The object stands this tall, and its drawing on the page is well under half of that, because rays that converge shrink whatever is far away. That shrinking is perspective, and it is exactly what your own eye does.","live":["view","heading_rays","page","page_tag","body","body_tag","eye","ray_hi","ray_lo","image","image_tag"],"does":[[60.087,"tall is shown on the screen, written out."],[61.120000000000005,"image is indicated — a transient flash."],[62.687000000000005,"verdict is shown on the screen, written out."]]},{"start":73.3655,"say":"Now change one thing only. 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At a slant, the drawing is called oblique. Square on, it is orthographic, and that includes the axonometric views, of which isometric is the one you have certainly seen.","live":["item_persp","heading_three"],"does":[[118.78800000000001,"item_obl is shown on the screen, written out."],[121.00600000000001,"item_orth is shown on the screen, written out."]]},{"start":128.955,"say":"Perspective, oblique, orthographic. One of the three shows the world as an eye receives it. 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Take two rails that really are parallel, and really are the same distance apart at every step of the way.","live":[],"does":[[229.35777083333335,"heading_vp is shown on the screen, written out."],[229.35777083333335,"road is shown on the screen, written out."],[232.15577083333335,"rail_left is shown on the screen, written out."],[232.3496485595913,"rail_right is shown on the screen, written out."]]},{"start":238.10827083333334,"say":"Their images do not stay apart. Every ray to the eye pulls them closer, and on the page they meet, at a single point on the horizon. That is the vanishing point, and nothing whatever in the world stands there.","live":["road","heading_vp","rail_left","rail_right"],"does":[[245.53777083333335,"skyline is shown on the screen, written out."],[247.56977083333334,"vanish is shown on the screen, written out."]]},{"start":251.95477083333333,"say":"Now lay sleepers across the track, equally spaced, one every metre. Out in the world every gap is the same. On the page they crowd together as they go back, and the last of them are almost on top of each other.","live":["road","heading_vp","rail_left","rail_right","skyline","vanish"],"does":[[252.8257708333333,"tie_1 is shown on the screen, written out."],[256.2967708333333,"tie_2 is shown on the screen, written out."],[259.0487708333333,"tie_3 is shown on the screen, written out."],[260.98777083333334,"tie_4 is shown on the screen, written out."],[262.07877083333335,"tie_5 is shown on the screen, written out."]]},{"start":266.0462708333333,"say":"So a perspective drawing is faithful to the eye and useless to a ruler. Two marks of the same length on the page can be a metre apart in the world, or a hundred, and the drawing will not tell you which.","live":["road","heading_vp","rail_left","rail_right","skyline","vanish","tie_1","tie_2","tie_3","tie_4","tie_5"],"does":[[269.6917708333333,"vp_note is shown on the screen, written out."],[272.2457708333333,"tie_2 is indicated — a transient flash."],[272.4657478858576,"tie_5 is indicated — a transient flash."]]},{"start":278.83677083333333,"say":"Which is why perspective belongs to painting, to photography, to film, and to the camera in every video game: all the places where the job is to look right. It is not where you go when the job is to be built.","live":["road","vp_note","heading_vp","rail_left","rail_right","skyline","vanish","tie_1","tie_2","tie_3","tie_4","tie_5"],"does":[[284.6067708333333,"vanish is indicated — a transient flash."],[292.71064583333333,"heading_vp is hidden from the screen — left the board."],[292.71064583333333,"road is hidden from the screen — left the board."],[292.71064583333333,"rail_left is hidden from the screen — road left the board."],[292.71064583333333,"rail_right is hidden from the screen — road left the board."],[292.71064583333333,"skyline is hidden from the screen — road left the board."],[292.71064583333333,"vanish is hidden from the screen — road left the board."],[292.71064583333333,"tie_1 is hidden from the screen — road left the board."],[292.71064583333333,"tie_2 is hidden from the screen — road left the board."],[292.71064583333333,"tie_3 is hidden from the screen — road left the board."],[292.71064583333333,"tie_4 is hidden from the screen — road left the board."],[292.71064583333333,"tie_5 is hidden from the screen — road left the board."],[292.71064583333333,"vp_note is hidden from the screen — left the board."]]}]},{"title":"Oblique: One True Face","start":293.75231249999996,"end":410.9680208333333,"objects":{"back_right":"a Line [yellow] drawn in cube (start=((2.7 + (1.4142135623730951 * kd)), (0.6 + (1.4142135623730951 …, end=((2.7 + (1.4142135623730951 * kd)), (2.6 + (1.4142135623730951 …)","back_top":"a Line [yellow] drawn in cube (start=((2.7 + (1.4142135623730951 * kd)), (2.6 + (1.4142135623730951 …, end=((0.7 + (1.4142135623730951 * kd)), (2.6 + (1.4142135623730951 …)","cube":"a Figure (x_range=(0.0, 5.2), y_range=(0.0, 5.0), aspect=(5.2, 5.0))","depth_1":"a Line [yellow] drawn in cube (start=(2.7, 0.6), end=((2.7 + (1.4142135623730951 * kd)), (0.6 + (1.4142135623730951 …)","depth_2":"a Line [yellow] drawn in cube (start=(2.7, 2.6), end=((2.7 + (1.4142135623730951 * kd)), (2.6 + (1.4142135623730951 …)","depth_3":"a Line [yellow] drawn in cube (start=(0.7, 2.6), end=((0.7 + (1.4142135623730951 * kd)), (2.6 + (1.4142135623730951 …)","depth_angle":"an Angle [magenta] labelled \"45 degree\" drawn in cube (vertex=(0.7, 2.6), sides=((2.7, 2.6), (1.4, 3.3)), radius=0.35)","edge_bottom":"a Line [blue] labelled \"a\" drawn in cube (start=(0.7, 0.6), end=(2.7, 0.6))","edge_left":"a Line [blue] drawn in cube (start=(0.7, 2.6), end=(0.7, 0.6))","edge_right":"a Line [blue] drawn in cube (start=(2.7, 0.6), end=(2.7, 2.6))","edge_top":"a Line [blue] drawn in cube (start=(2.7, 2.6), end=(0.7, 2.6))","front_face":"a Line [green] drawn in plan (start=(1.2, 0.7), end=(1.2, 1.9))","heading_cube":"a Heading that says \"Oblique: The Front Face Is True\"","heading_slant":"a Heading that says \"Parallel Rays, Struck at a Slant\"","k_cab":"a Math [text] that says \"$upright(\"cabinet:\") thin k = frac(1, 2)$\"","k_cav":"a Math [text] that says \"$upright(\"cavalier:\") thin k = 1$\"","kd":"a VariableNumber (initial_value=1.0)","oblique_def":"a Panel that says \"Parallel rays, striking the page at a slant. The face lying in the page is drawn exactly: true lengths, true angles. Depth is drawn at a chosen angle and a chosen fraction of its true length.\"","paper":"a Line [gray] drawn in plan (start=(1.2, 0.3), end=(1.2, 3.8))","plan":"a Figure (x_range=(0.0, 4.2), y_range=(0.0, 4.0), aspect=(4.2, 4.0))","rear_face":"a Line [green] drawn in plan (start=(1.2, 2.3099999999999996), end=(1.2, 3.51))","slant_angle":"an Angle [magenta] labelled \"alpha\" drawn in plan (vertex=(1.2, 2.3099999999999996), sides=((1.2, 1.9), (2.6, 0.7)))","slant_hi":"a Line [yellow] drawn in plan (start=(2.6, 1.9), end=(1.2, 3.51))","slant_lo":"a Line [yellow] drawn in plan (start=(2.6, 0.7), end=(1.2, 2.3099999999999996))","slant_mid":"a Line [yellow] drawn in plan (start=(2.6, 1.3), end=(1.2, 2.91))","slant_note":"a Tex [text] that says \"Nothing converges, so nothing shrinks with distance.\"","solid_plan":"a Polygon [blue] drawn in plan (vertices=((1.2, 0.7), (2.6, 0.7), (2.6, 1.9), (1.2, 1.9)), fill_opacity=0.22)"},"beats":[{"start":293.75231249999996,"say":"Cross into the parallel family. Here is the page again, edge on, and the object behind it, and this time the rays arrive at a slant.","live":[],"does":[[293.75231249999996,"heading_slant is shown on the screen, written out."],[293.75231249999996,"plan is shown on the screen, written out."],[296.96831249999997,"paper is shown on the screen, written out."],[299.53431249999994,"solid_plan is shown on the screen, written out."],[302.7733125,"slant_lo is shown on the screen, written out."],[303.1758771950092,"slant_mid is shown on the screen, written out."],[303.8347335526315,"slant_hi is shown on the screen, written out."]]},{"start":304.29031249999997,"say":"Set the front face of the object flat against the page. Those points are already on it, so they draw themselves: same size, same shape, same angles, exactly.","live":["plan","heading_slant","paper","solid_plan","slant_lo","slant_mid","slant_hi"],"does":[[310.22331249999996,"front_face is shown on the screen, written out."]]},{"start":315.16531249999997,"say":"Now follow the rear face. Its rays are slanted, so they land further along the page, and every one of its points is shifted by the very same amount, because the rays never converge.","live":["plan","heading_slant","paper","solid_plan","slant_lo","slant_mid","slant_hi","front_face"],"does":[[322.3863125,"rear_face is shown on the screen, written out."],[325.1963125,"slant_angle is shown on the screen, written out."]]},{"start":326.78331249999997,"say":"That shift is the depth of the drawing, and the slant angle alone decides how big it is. Near and far are drawn at one single scale.","live":["plan","heading_slant","paper","solid_plan","slant_lo","slant_mid","slant_hi","front_face","rear_face","slant_angle"],"does":[[330.18531249999995,"slant_angle is indicated — a transient flash."],[333.0763125,"slant_note is shown on the screen, written out."],[335.9553125,"heading_slant is hidden from the screen — left the board."],[335.9553125,"plan is hidden from the screen — left the board."],[335.9553125,"paper is hidden from the screen — plan left the board."],[335.9553125,"solid_plan is hidden from the screen — plan left the board."],[335.9553125,"slant_lo is hidden from the screen — plan left the board."],[335.9553125,"slant_mid is hidden from the screen — plan left the board."],[335.9553125,"slant_hi is hidden from the screen — plan left the board."],[335.9553125,"front_face is hidden from the screen — plan left the board."],[335.9553125,"rear_face is hidden from the screen — plan left the board."],[335.9553125,"slant_angle is hidden from the screen — plan left the board."],[335.9553125,"slant_note is hidden from the screen — left the board."]]},{"start":336.55531249999996,"say":"Here is what that gives you on a cube. The front face is a true square: right angles, equal sides. Measure it on the page and you have measured the object.","live":[],"does":[[336.55531249999996,"heading_cube is shown on the screen, written out."],[336.55531249999996,"cube is shown on the screen, written out."],[340.52631249999996,"edge_bottom is shown on the screen, written out."],[340.7515116150442,"edge_right is shown on the screen, written out."],[341.2019098451327,"edge_top is shown on the screen, written out."],[341.6048091814159,"edge_left is shown on the screen, written out."]]},{"start":347.26731249999995,"say":"The depth runs off at forty five degrees, the same length from every corner, and the far face is a second true square, parallel to the first.","live":["cube","heading_cube","edge_bottom","edge_right","edge_top","edge_left"],"does":[[347.74331249999994,"depth_1 is shown on the screen, written out."],[347.85851060403496,"depth_2 is shown on the screen, written out."],[348.1088644688867,"depth_3 is shown on the screen, written out."],[349.42731249999997,"depth_angle is shown on the screen, written out."],[352.41131249999995,"back_right is shown on the screen, written out."],[352.68675732255707,"back_top is shown on the screen, written out."]]},{"start":356.54031249999997,"say":"Drawn at full depth, this is cavalier oblique, and every single edge on the page is a true length. But look at it: it seems to swell as it goes back, because your eye expects far things to be smaller, and this drawing refuses.","live":["cube","heading_cube","edge_bottom","edge_right","edge_top","edge_left","depth_1","depth_2","depth_3","depth_angle","back_right","back_top"],"does":[[358.5833124999999,"cube moves to a new place on the board."],[358.5833124999999,"k_cav is shown on the screen, written out."]]},{"start":371.5948125,"say":"So draughtsmen halve the depth. That is cabinet oblique: the front face still exactly true, the receding edges at half scale, and the solid suddenly looks like a solid.","live":["k_cav","cube","heading_cube","edge_bottom","edge_right","edge_top","edge_left","depth_1","depth_2","depth_3","depth_angle","back_right","back_top"],"does":[[372.7323125,"depth_1 is redrawn as the numbers it depends on change."],[372.7323125,"depth_2 is redrawn as the numbers it depends on change."],[372.7323125,"depth_3 is redrawn as the numbers it depends on change."],[372.7323125,"back_right is redrawn as the numbers it depends on change."],[372.7323125,"back_top is redrawn as the numbers it depends on change."],[372.7323125,"kd ticks to 0.5."],[374.71831249999997,"k_cab is shown on the screen, written out."]]},{"start":384.2228125,"say":"The price is easy to state. Only the face lying in the page can be measured directly, and only lines parallel to that face keep their length. Everything in the depth direction is a convention rather than a measurement.","live":["k_cav","k_cab","cube","heading_cube","edge_bottom","edge_right","edge_top","edge_left","depth_1","depth_2","depth_3","depth_angle","back_right","back_top"],"does":[[384.73331249999995,"oblique_def is shown on the screen, written out."],[388.86631249999994,"oblique_def (the \"lying in the page\" part) is emphasized."],[397.7948125,"oblique_def (the \"lying in the page\" part) is no longer emphasized."]]},{"start":398.39481249999994,"say":"It is cheap to draw by hand and it reads at a glance, which is why oblique filled workshop notebooks and furniture drawings for two hundred years. The cabinet convention is named after cabinets.","live":["oblique_def","k_cav","k_cab","cube","heading_cube","edge_bottom","edge_right","edge_top","edge_left","depth_1","depth_2","depth_3","depth_angle","back_right","back_top"],"does":[[407.15931249999994,"k_cab is indicated — a transient flash."],[409.9263541666666,"cube is hidden from the screen — left the board."],[409.9263541666666,"edge_bottom is hidden from the screen — cube left the board."],[409.9263541666666,"edge_right is hidden from the screen — cube left the board."],[409.9263541666666,"edge_top is hidden from the screen — cube left the board."],[409.9263541666666,"edge_left is hidden from the screen — cube left the board."],[409.9263541666666,"depth_1 is hidden from the screen — cube left the board."],[409.9263541666666,"depth_2 is hidden from the screen — cube left the board."],[409.9263541666666,"depth_3 is hidden from the screen — cube left the board."],[409.9263541666666,"depth_angle is hidden from the screen — cube left the board."],[409.9263541666666,"back_right is hidden from the screen — cube left the board."],[409.9263541666666,"back_top is hidden from the screen — cube left the board."],[409.9263541666666,"heading_cube is hidden from the screen — left the board."],[409.9263541666666,"k_cab is hidden from the screen — left the board."],[409.9263541666666,"k_cav is hidden from the screen — left the board."],[409.9263541666666,"oblique_def is hidden from the screen — left the board."]]}]},{"title":"Orthographic and Axonometric","start":410.9680208333333,"end":560.365,"objects":{"axo_def":"a Panel that says \"Parallel rays, square on to the page, with the object turned so all three axes show. Parallel edges stay parallel, so nothing shrinks with distance.\"","block":"a Solid [blue] drawn in solid_space (upper=<function>, x_range=(0.3, 2.3), y_range=(0.3, 1.9))","front_face":"a Line [green] drawn in plan (start=(1.2, 0.7), end=(1.2, 1.9))","front_tag":"a Math [text] that says \"$upright(\"front\")$\" drawn in views","front_view":"a Polygon [blue] drawn in views (vertices=((0.6, 0.5), (2.6, 0.5), (2.6, 1.9), (0.6, 1.9)), fill_opacity=0.2)","heading_axo":"a Heading that says \"Turn the Object Instead\"","heading_square":"a Heading that says \"Rays Square On to the Page\"","heading_views":"a Heading that says \"One View Is Not Enough\"","iso_angle":"a Math [text] that says \"$120 degree quad upright(\"between the drawn axes\")$\"","iso_scale":"a Math [text] that says \"$0.816 quad upright(\"along every axis\")$\"","line":"a Line [blue] drawn in solid_space (start=(0.0, 0.0, 0.0), end=(2.0, 0.0, 0.0), dashed=True)","line_2":"a Line [blue] drawn in solid_space (start=(2.0, 0.0, 0.0), end=(2.0, 1.6, 0.0), dashed=True)","line_3":"a Line [blue] drawn in solid_space (start=(2.0, 1.6, 0.0), end=(2.0, 1.6, 1.4), dashed=True)","link_high":"a Line [gray] drawn in views (start=(2.6, 1.9), end=(3.1, 1.9), dashed=True)","link_left":"a Line [gray] drawn in views (start=(0.6, 1.9), end=(0.6, 2.4), dashed=True)","link_low":"a Line [gray] drawn in views (start=(2.6, 0.5), end=(3.1, 0.5), dashed=True)","link_right":"a Line [gray] drawn in views (start=(2.6, 1.9), end=(2.6, 2.4), dashed=True)","ortho_note":"a Tex [text] that says \"The depth has left the page, and what is left is exact.\"","paper":"a Line [gray] drawn in plan (start=(1.2, 0.3), end=(1.2, 3.8))","plan":"a Figure (x_range=(0.0, 4.2), y_range=(0.0, 4.0), aspect=(4.2, 4.0))","plate":"a Polygon [gray] drawn in plan (vertices=((1.2, 0.7), (1.5, 0.7), (1.5, 1.9), (1.2, 1.9)), fill_opacity=0.35)","ray_hi":"a Line [yellow] drawn in plan (start=(2.6, 1.9), end=(1.2, (1.9 + (slope * 1.4))))","ray_lo":"a Line [yellow] drawn in plan (start=(2.6, 0.7), end=(1.2, (0.7 + (slope * 1.4))))","rear_face":"a Line [red] drawn in plan (start=(1.2, (0.7 + (slope * 1.4))), end=(1.2, (1.9 + (slope * 1.4))))","side_tag":"a Math [text] that says \"$upright(\"side\")$\" drawn in views","side_view":"a Polygon [blue] drawn in views (vertices=((3.1, 0.5), (4.3, 0.5), (4.3, 1.9), (3.1, 1.9)), fill_opacity=0.2)","slope":"a VariableNumber (initial_value=1.15)","solid_plan":"a Polygon [blue] drawn in plan (vertices=((1.2, 0.7), (2.6, 0.7), (2.6, 1.9), (1.2, 1.9)), fill_opacity=0.22)","solid_space":"an Axes3D (x_range=(0.0, 2.6), y_range=(0.0, 2.2), z_range=(0.0, 1.8))","square_mark":"an Angle [magenta] drawn in plan (vertex=(1.2, 0.7), sides=((1.2, 1.9), (2.6, 0.7)), right_angle=True)","top_tag":"a Math [text] that says \"$upright(\"top\")$\" drawn in views","top_view":"a Polygon [blue] drawn in views (vertices=((0.6, 2.4), (2.6, 2.4), (2.6, 3.6), (0.6, 3.6)), fill_opacity=0.2)","views":"a Figure (x_range=(0.0, 5.2), y_range=(0.0, 4.2), aspect=(5.2, 4.2))","views_note":"a Tex [text] that says \"Three flat views, every one of them measurable.\""},"beats":[{"start":410.9680208333333,"say":"One change is left to make. Here are the slanted rays again, with the front face true on the page, and the rear face shifted along it.","live":[],"does":[[410.9680208333333,"heading_square is shown on the screen, written out."],[410.9680208333333,"plan is shown on the screen, written out."],[413.0000208333333,"paper is shown on the screen, written out."],[413.0000208333333,"solid_plan is shown on the screen, written out."],[413.4530208333333,"ray_lo is shown on the screen, written out."],[413.6433014521009,"ray_hi is shown on the screen, written out."],[414.9150208333333,"front_face is shown on the screen, written out."],[417.3530208333333,"rear_face is shown on the screen, written out."]]},{"start":419.1955208333333,"say":"Now swing the rays square to the page, and watch the shift close up. There it goes: the rear face lands exactly on top of the front face, and the depth of the object has gone out of the drawing altogether.","live":["plan","heading_square","paper","solid_plan","ray_lo","ray_hi","front_face","rear_face"],"does":[[419.8460208333333,"ray_lo is redrawn as the numbers it depends on change."],[419.8460208333333,"ray_hi is redrawn as the numbers it depends on change."],[419.8460208333333,"rear_face is redrawn as the numbers it depends on change."],[419.8460208333333,"slope ticks to 0.0."],[426.7300208333333,"square_mark is shown on the screen, written out."],[430.2950208333333,"ortho_note is shown on the screen, written out."]]},{"start":432.9030208333333,"say":"This is orthographic projection, and losing the depth is not a defect. What is left on the page is one face, dead true: true lengths, true angles, and every one of them measurable.","live":["plan","ortho_note","heading_square","paper","solid_plan","ray_lo","ray_hi","front_face","rear_face","square_mark"],"does":[[439.7990208333333,"rear_face is indicated — a transient flash."]]},{"start":446.0765208333333,"say":"But one view like this cannot describe a solid. A thin plate and a long block give exactly the same front view, so the answer is to draw more than one of them.","live":null,"does":[[450.3140208333333,"plate is shown on the screen, written out."],[454.0290208333333,"plate is hidden from the screen."],[456.3055208333333,"heading_square is hidden from the screen — left the board."],[456.3055208333333,"ortho_note is hidden from the screen — left the board."],[456.3055208333333,"plan is hidden from the screen — left the board."],[456.3055208333333,"paper is hidden from the screen — plan left the board."],[456.3055208333333,"solid_plan is hidden from the screen — plan left the board."],[456.3055208333333,"ray_lo is hidden from the screen — plan left the board."],[456.3055208333333,"ray_hi is hidden from the screen — plan left the board."],[456.3055208333333,"front_face is hidden from the screen — plan left the board."],[456.3055208333333,"rear_face is hidden from the screen — plan left the board."],[456.3055208333333,"square_mark is hidden from the screen — plan left the board."]]},{"start":456.9055208333333,"say":"So here is the standard set of three. The front view, straight on. Above it the top view, looking down from over the block. Beside it the side view, looking in from the right.","live":[],"does":[[456.9055208333333,"heading_views is shown on the screen, written out."],[456.9055208333333,"views is shown on the screen, written out."],[459.9240208333333,"front_view is shown on the screen, written out."],[459.9240208333333,"front_tag is shown on the screen, written out."],[462.4200208333333,"top_view is shown on the screen, written out."],[462.4200208333333,"top_tag is shown on the screen, written out."],[465.8450208333333,"side_view is shown on the screen, written out."],[465.8450208333333,"side_tag is shown on the screen, written out."]]},{"start":468.5925208333333,"say":"And the three are locked together. Width is shared by the front and the top, so those two line up along here. Height is shared by the front and the side, so they line up across here. Between them, three flat drawings hold every dimension of the block, each one at true size.","live":["views","heading_views","front_view","front_tag","top_view","top_tag","side_view","side_tag"],"does":[[474.3630208333333,"link_left is shown on the screen, written out."],[474.3630208333333,"link_right is shown on the screen, written out."],[478.5310208333333,"link_low is shown on the screen, written out."],[478.5310208333333,"link_high is shown on the screen, written out."],[484.9050208333333,"views_note is shown on the screen, written out."]]},{"start":486.3985208333333,"say":"Nothing here looks like the object, and that is the whole trade. The drawing has stopped being a picture and become a measurement. This is the language that workshop drawings and building drawings are written in.","live":["views","views_note","heading_views","front_view","front_tag","top_view","top_tag","side_view","side_tag","link_left","link_right","link_low","link_high"],"does":[[492.69102083333325,"front_view is indicated — a transient flash."],[493.03607186200657,"top_view is indicated — a transient flash."],[493.7261739193531,"side_view is indicated — a transient flash."],[498.2060208333333,"heading_views is hidden from the screen — left the board."],[498.2060208333333,"views is hidden from the screen — left the board."],[498.2060208333333,"front_view is hidden from the screen — views left the board."],[498.2060208333333,"front_tag is hidden from the screen — views left the board."],[498.2060208333333,"top_view is hidden from the screen — views left the board."],[498.2060208333333,"top_tag is hidden from the screen — views left the board."],[498.2060208333333,"side_view is hidden from the screen — views left the board."],[498.2060208333333,"side_tag is hidden from the screen — views left the board."],[498.2060208333333,"link_left is hidden from the screen — views left the board."],[498.2060208333333,"link_right is hidden from the screen — views left the board."],[498.2060208333333,"link_low is hidden from the screen — views left the board."],[498.2060208333333,"link_high is hidden from the screen — views left the board."],[498.2060208333333,"views_note is hidden from the screen — left the board."]]},{"start":499.4060208333333,"say":"There is a middle way, and you have seen it a thousand times. Keep the rays parallel and square to the page, exactly as they are now, but turn the block itself, so that three faces show at once.","live":[],"does":[[499.4060208333333,"heading_axo is shown on the screen, written out."],[499.4060208333333,"solid_space is shown on the screen, written out."],[500.25302083333327,"block is shown on the screen, written out."],[507.97402083333327,"solid_space turns to a new orientation in its own slot."]]},{"start":511.91702083333325,"say":"Now it reads as a solid again. But look along its edges: the ones that are parallel on the block are still parallel on the page. Nothing converges, so this is still a parallel projection, and it can still be measured.","live":["solid_space","heading_axo","block"],"does":[[521.1590208333332,"solid_space moves to a new place on the board."],[521.1590208333332,"axo_def is shown on the screen, written out."]]},{"start":526.4380208333333,"say":"Choose the tilt so that the three axes are treated alike, and you have the isometric view. On the page those axes stand a hundred and twenty degrees apart, and all three are shortened by the very same factor, about zero point eight one six.","live":["axo_def","solid_space","heading_axo","block"],"does":[[533.4970208333333,"line is shown on the screen, drawn."],[533.4970208333333,"line_2 is shown on the screen, drawn."],[533.4970208333333,"line_3 is shown on the screen, drawn."],[534.6690208333333,"iso_angle is shown on the screen, written out."],[535.6067715235879,"line is hidden from the screen."],[535.6067715235879,"line_2 is hidden from the screen."],[535.6067715235879,"line_3 is hidden from the screen."],[538.0590208333333,"iso_scale is shown on the screen, written out."]]},{"start":541.7130208333333,"say":"That is the pair of properties that made it famous. It looks like a solid, and one ruler measures all three directions at once. Which is why isometric is the language of assembly instructions, of pipe runs, of patent drawings, and of a whole generation of video games.","live":["axo_def","iso_angle","iso_scale","solid_space","heading_axo","block"],"does":[[547.1580208333332,"iso_scale is indicated — a transient flash."],[559.3233333333333,"axo_def is hidden from the screen — left the board."],[559.3233333333333,"heading_axo is hidden from the screen — left the board."],[559.3233333333333,"iso_angle is hidden from the screen — left the board."],[559.3233333333333,"iso_scale is hidden from the screen — left the board."],[559.3233333333333,"solid_space is hidden from the screen — left the board."],[559.3233333333333,"block is hidden from the screen — solid_space left the board."]]}]},{"title":"Which One Is Real?","start":560.365,"end":704.0419791666666,"objects":{"closing_note":"a Text [text] that says \"Nothing here is a lie. Each system throws something different away, and each is chosen for what it keeps.\"","flat_dn":"an Arrow [yellow] drawn in limit (start=(8.6, 1.1), end=(4.5, 1.1))","flat_up":"an Arrow [yellow] drawn in limit (start=(8.6, 2.9), end=(4.5, 2.9))","heading_limit":"a Heading that says \"An Eye Infinitely Far Away\"","heading_uses":"a Heading that says \"What Each One Keeps\"","limit":"a Figure (x_range=(0.0, 8.8), y_range=(0.2, 4.0), aspect=(8.8, 3.8))","limit_note":"a Tex [text] that says \"A parallel projection is a perspective from infinitely far away.\"","mark":"a Line [green] drawn in limit (start=(4.5, (2.0 - spread)), end=(4.5, (2.0 + spread)))","ray_dn":"a Line [yellow] drawn in limit (start=(7.5, 1.1), end=(<VariableNumber station = 0.2>, 2.0))","ray_up":"a Line [yellow] drawn in limit (start=(7.5, 2.9), end=(<VariableNumber station = 0.2>, 2.0))","sheet":"a Line [gray] drawn in limit (start=(4.5, 0.6), end=(4.5, 3.6))","spread":"a VariableNumber (initial_value=0.225)","station":"a VariableNumber (initial_value=3.5)","table":"a Table [text] that says \"System What it keeps Where it is used Perspective how the world looks art, photography, film Oblique one true face sketches, furniture, notebooks Orthographic true sizes, view by view workshop and site drawings Isometric one scale, all thr…\" (rows=(('System', 'What it keeps', 'Where it is used'), ('Perspective…, header=True)","thing":"a Polygon [blue] drawn in limit (vertices=((7.5, 1.1), (8.3, 1.1), (8.3, 2.9), (7.5, 2.9)), fill_opacity=0.22)","watcher":"a Point [red] labelled \"E\" drawn in limit (location=(<VariableNumber station = 0.2>, 2.0))"},"beats":[{"start":560.365,"say":"So which of these is real? There is one honest answer, and then a twist. Perspective is what an eye receives, because an eye has a single pupil, and every ray that reaches it converges there. A photograph is a perspective projection whether the photographer wanted one or not.","live":[],"does":[[560.365,"heading_limit is shown on the screen, written out."],[560.365,"limit is shown on the screen, written out."],[563.802,"sheet is shown on the screen, written out."],[563.802,"thing is shown on the screen, written out."],[569.711,"watcher is shown on the screen, written out."],[571.836,"ray_up is shown on the screen, written out."],[572.0323046456595,"ray_dn is shown on the screen, written out."],[575.214,"mark is shown on the screen, written out."]]},{"start":578.844,"say":"Now the twist. Carry the eye backwards, and leave the object and the page exactly where they are. The rays flatten out. The image on the page grows, and the near and far parts of the object are treated more and more alike.","live":["limit","heading_limit","sheet","thing","watcher","ray_up","ray_dn","mark"],"does":[[581.468,"watcher is redrawn as the numbers it depends on change."],[581.468,"ray_up is redrawn as the numbers it depends on change."],[581.468,"ray_dn is redrawn as the numbers it depends on change."],[581.468,"mark is redrawn as the numbers it depends on change."],[581.468,"station ticks to 0.2."],[581.468,"spread ticks to 0.53."]]},{"start":594.2235000000001,"say":"Take that to its end. Put the eye infinitely far away, and the rays are exactly parallel. The shrinking stops altogether, and the mark on the page is precisely the size of the object.","live":null,"does":[[597.01,"watcher is hidden from the screen."],[597.01,"ray_up is hidden from the screen."],[597.01,"ray_dn is hidden from the screen."],[599.773,"flat_up is shown on the screen, written out."],[600.3858732183046,"flat_dn is shown on the screen, written out."],[602.176,"mark is redrawn as the numbers it depends on change."],[602.176,"spread ticks to 0.9."],[604.753,"limit_note is shown on the screen, written out."]]},{"start":607.7335,"say":"So a parallel projection is not a mistake about vision. It is the view from an infinite distance, which no eye can ever occupy. That is why an oblique or an isometric drawing always looks very slightly impossible. It is impossible on purpose.","live":["limit","limit_note","heading_limit","sheet","thing","mark","flat_up","flat_dn"],"does":[[612.726,"mark is indicated — a transient flash."],[612.726,"thing is indicated — a transient flash."],[623.953,"heading_limit is hidden from the screen — left the board."],[623.953,"limit is hidden from the screen — left the board."],[623.953,"sheet is hidden from the screen — limit left the board."],[623.953,"thing is hidden from the screen — limit left the board."],[623.953,"mark is hidden from the screen — limit left the board."],[623.953,"flat_up is hidden from the screen — limit left the board."],[623.953,"flat_dn is hidden from the screen — limit left the board."],[623.953,"limit_note is hidden from the screen — left the board."]]},{"start":624.553,"say":"Then the useful question is not which one is real. It is what each one throws away, and who can afford to lose it.","live":[],"does":[[624.553,"heading_uses is shown on the screen, written out."],[625.83,"table is shown on the screen, written out."]]},{"start":632.0375,"say":"Perspective keeps the look of the world and throws measurement away. For a painting, a photograph, a film, or the camera in a game engine, that is exactly the right bargain.","live":["heading_uses"],"does":[[632.444,"table is shown on the screen, written out."],[642.602,"table (the \"row=2\" part) is indicated — a transient flash."]]},{"start":644.05,"say":"Oblique keeps one true face and throws the rest away. It is the quickest useful pictorial drawing there is, which is why it filled notebooks and furniture drawings for two hundred years.","live":null,"does":[[644.398,"table is shown on the screen, written out."],[648.787,"table (the \"row=3\" part) is indicated — a transient flash."]]},{"start":655.818,"say":"Orthographic throws the picture away entirely and keeps every size. Nothing is ever built from a perspective drawing. Buildings and machine parts are built from orthographic views.","live":null,"does":[[656.166,"table is shown on the screen, written out."],[659.742,"table (the \"row=4\" part) is indicated — a transient flash."]]},{"start":668.0285,"say":"And isometric sits between the two. One scale for all three axes, a solid that reads as a solid, and a drawing you can still measure: assembly guides, pipe runs, and a great deal of game art.","live":null,"does":[[668.702,"table is shown on the screen, written out."],[676.655,"table (the \"row=5\" part) is indicated — a transient flash."]]},{"start":681.69,"say":"So there is the whole map, and one question decides it. What is the drawing for?","live":null,"does":[[683.0830000000001,"closing_note is shown on the screen, written out."]]},{"start":687.967,"say":"If it must look right, let the rays converge. If it must be measured, keep them parallel, and then choose what stays true: one face for oblique, one view at a time for orthographic, all three axes alike for isometric.","live":["closing_note","heading_uses"],"does":[[697.022,"table (the \"row=3\" part) is indicated — a transient flash."],[698.961,"table (the \"row=4\" part) is indicated — a transient flash."],[701.701,"table (the \"row=5\" part) is indicated — a transient flash."],[703.0003125000001,"closing_note is hidden from the screen — left the board."],[703.0003125000001,"heading_uses is hidden from the screen — left the board."],[703.0003125000001,"table is hidden from the screen — left the board."]]}]}]},"durationSeconds":704,"chapters":[{"title":"From a Solid to a Flat Page","startSeconds":0,"narration":"Here is a solid block. It has width, it has depth, and it has height: three directions at once, and six flat faces. And here is the difficulty. A page is flat. It has two directions to spend, and the block has three, so one of those three cannot be drawn as it truly is. Every drawing system in this lecture is a different answer to that. So look down on the whole business from above. The page stands edge on, as this pale vertical line. The object sits over here, and the drawing will appear where the rays cross it. Suppose every ray runs to one single point, an eye out at the front. From each corner of the object, a straight ray to that point, and a mark where it crosses. Between the two marks is the drawing, and its height is h prime. Now compare the two heights. The object stands this tall, and its drawing on the page is well under half of that, because rays that converge shrink whatever is far away. That shrinking is perspective, and it is exactly what your own eye does. Now change one thing only. Take the eye away, and let every ray run parallel to its neighbours instead of gathering to a point. Nothing converges now, so nothing shrinks. The drawing grows until it is exactly as tall as the object itself, and a ruler laid on the page measures the real thing. So there are two great families of projection, and every drawing ever made belongs to one of them. Rays that meet at a point, or rays that never meet at all. The converging family has just one member, and it is called perspective. The parallel family splits in two, by the angle at which the rays strike the page. At a slant, the drawing is called oblique. Square on, it is orthographic, and that includes the axonometric views, of which isometric is the one you have certainly seen. Perspective, oblique, orthographic. One of the three shows the world as an eye receives it. The other two show something no eye can ever see, and that is exactly why they are what you build from."},{"title":"Perspective: What the Eye Receives","startSeconds":144.40677083333333,"narration":"Start with the converging family. One eye, one page, one post. The eye is here, and the page stands at a fixed distance in front of it, which I will call f. The post stands out at a distance Z, and it is h tall. Now the two rays. One from its top to the eye, one from its foot to the eye. Each crosses the page on the way, and between the two crossings is the drawn image, of height h prime. Now the geometry does the work. The big triangle from the eye out to the post, and the small one from the eye to the page, share an apex and a direction, so they are similar. Similar triangles give a proportion: h prime over h equals f over Z. Rearrange that, and the drawn height is f times h over Z. There is the whole of perspective in a single line: what you draw is inversely proportional to how far away the thing is. Put numbers on that. At one page distance out, the drawing is life size. At twice the distance, half size. At four times out, a quarter. And here it is happening. The post walks out, and its mark on the page closes down. Further still, and smaller again. Bring it back in, and the mark swells. Now the famous consequence. Take two rails that really are parallel, and really are the same distance apart at every step of the way. Their images do not stay apart. Every ray to the eye pulls them closer, and on the page they meet, at a single point on the horizon. That is the vanishing point, and nothing whatever in the world stands there. Now lay sleepers across the track, equally spaced, one every metre. Out in the world every gap is the same. On the page they crowd together as they go back, and the last of them are almost on top of each other. So a perspective drawing is faithful to the eye and useless to a ruler. Two marks of the same length on the page can be a metre apart in the world, or a hundred, and the drawing will not tell you which. Which is why perspective belongs to painting, to photography, to film, and to the camera in every video game: all the places where the job is to look right. It is not where you go when the job is to be built."},{"title":"Oblique: One True Face","startSeconds":293.75231249999996,"narration":"Cross into the parallel family. Here is the page again, edge on, and the object behind it, and this time the rays arrive at a slant. Set the front face of the object flat against the page. Those points are already on it, so they draw themselves: same size, same shape, same angles, exactly. Now follow the rear face. Its rays are slanted, so they land further along the page, and every one of its points is shifted by the very same amount, because the rays never converge. That shift is the depth of the drawing, and the slant angle alone decides how big it is. Near and far are drawn at one single scale. Here is what that gives you on a cube. The front face is a true square: right angles, equal sides. Measure it on the page and you have measured the object. The depth runs off at forty five degrees, the same length from every corner, and the far face is a second true square, parallel to the first. Drawn at full depth, this is cavalier oblique, and every single edge on the page is a true length. But look at it: it seems to swell as it goes back, because your eye expects far things to be smaller, and this drawing refuses. So draughtsmen halve the depth. That is cabinet oblique: the front face still exactly true, the receding edges at half scale, and the solid suddenly looks like a solid. The price is easy to state. Only the face lying in the page can be measured directly, and only lines parallel to that face keep their length. Everything in the depth direction is a convention rather than a measurement. It is cheap to draw by hand and it reads at a glance, which is why oblique filled workshop notebooks and furniture drawings for two hundred years. The cabinet convention is named after cabinets."},{"title":"Orthographic and Axonometric","startSeconds":410.9680208333333,"narration":"One change is left to make. Here are the slanted rays again, with the front face true on the page, and the rear face shifted along it. Now swing the rays square to the page, and watch the shift close up. There it goes: the rear face lands exactly on top of the front face, and the depth of the object has gone out of the drawing altogether. This is orthographic projection, and losing the depth is not a defect. What is left on the page is one face, dead true: true lengths, true angles, and every one of them measurable. But one view like this cannot describe a solid. A thin plate and a long block give exactly the same front view, so the answer is to draw more than one of them. So here is the standard set of three. The front view, straight on. Above it the top view, looking down from over the block. Beside it the side view, looking in from the right. And the three are locked together. Width is shared by the front and the top, so those two line up along here. Height is shared by the front and the side, so they line up across here. Between them, three flat drawings hold every dimension of the block, each one at true size. Nothing here looks like the object, and that is the whole trade. The drawing has stopped being a picture and become a measurement. This is the language that workshop drawings and building drawings are written in. There is a middle way, and you have seen it a thousand times. Keep the rays parallel and square to the page, exactly as they are now, but turn the block itself, so that three faces show at once. Now it reads as a solid again. But look along its edges: the ones that are parallel on the block are still parallel on the page. Nothing converges, so this is still a parallel projection, and it can still be measured. Choose the tilt so that the three axes are treated alike, and you have the isometric view. On the page those axes stand a hundred and twenty degrees apart, and all three are shortened by the very same factor, about zero point eight one six. That is the pair of properties that made it famous. It looks like a solid, and one ruler measures all three directions at once. Which is why isometric is the language of assembly instructions, of pipe runs, of patent drawings, and of a whole generation of video games."},{"title":"Which One Is Real?","startSeconds":560.365,"narration":"So which of these is real? There is one honest answer, and then a twist. Perspective is what an eye receives, because an eye has a single pupil, and every ray that reaches it converges there. A photograph is a perspective projection whether the photographer wanted one or not. Now the twist. Carry the eye backwards, and leave the object and the page exactly where they are. The rays flatten out. The image on the page grows, and the near and far parts of the object are treated more and more alike. Take that to its end. Put the eye infinitely far away, and the rays are exactly parallel. The shrinking stops altogether, and the mark on the page is precisely the size of the object. So a parallel projection is not a mistake about vision. It is the view from an infinite distance, which no eye can ever occupy. That is why an oblique or an isometric drawing always looks very slightly impossible. It is impossible on purpose. Then the useful question is not which one is real. It is what each one throws away, and who can afford to lose it. Perspective keeps the look of the world and throws measurement away. For a painting, a photograph, a film, or the camera in a game engine, that is exactly the right bargain. Oblique keeps one true face and throws the rest away. It is the quickest useful pictorial drawing there is, which is why it filled notebooks and furniture drawings for two hundred years. Orthographic throws the picture away entirely and keeps every size. Nothing is ever built from a perspective drawing. Buildings and machine parts are built from orthographic views. And isometric sits between the two. One scale for all three axes, a solid that reads as a solid, and a drawing you can still measure: assembly guides, pipe runs, and a great deal of game art. So there is the whole map, and one question decides it. What is the drawing for? If it must look right, let the rays converge. If it must be measured, keep them parallel, and then choose what stays true: one face for oblique, one view at a time for orthographic, all three axes alike for isometric."}]}}