Plot projection drawing spacing from a fixed viewpoint

Projection drawing spacing changes when points at equal depth intervals are mapped along rays from one viewpoint. In this original paper exercise, four object points sit equally far apart along a horizontal row. Their marks on a nearer vertical image line have unequal gaps. You can derive each mark before deciding where to draw it.

The example defines an ideal two-dimensional diagram. Its coordinates have abstract drawing units, with no centimetre or inch scale assigned to a real object. It supplies no camera model, lens setting, optical calibration or physical measurement result. Follow the declared rays and lines as a mathematical drawing exercise.

Put the viewpoint and image line on the page

Name the fixed viewpoint O and give it coordinates (0, 0). Let x increase to the right and y increase upward. Draw the image line at x = 2. It is vertical: every point on it has horizontal coordinate two, while its height can vary.

Place the four object points at (4, 4), (6, 4), (8, 4) and (10, 4). Call them A, B, C and D in that order. They all have height four. Their successive horizontal differences are two, two and two, so they occupy equally spaced positions along the object row.

Keep the object row separate from the image line. A is an object point at depth four; its projected mark will lie at depth two. They are different points in the drawing. Use a small label beside each mark so a later comparison does not accidentally treat a projected height as an object depth.

The image line sits between O and every object point. That is part of this example's arrangement. There is no reflecting surface, blocker or boundary test to solve. A ray is simply the straight route starting at O and continuing through its named object point.

Find the first mark from its own ray

Start with A = (4, 4). On the straight segment from O to A, halfway across the horizontal distance is also halfway up the vertical distance. At x = 2, the height is therefore two. Mark A's image position at (2, 2).

You can write the same calculation using a fraction t of the route from O. A point that fraction of the way toward A has coordinates (4t, 4t). The image line requires 4t = 2, giving t = 1/2. Substituting that fraction into the height gives 4 times one-half, or two.

This fraction describes a location on one stipulated straight segment. It is not a time, speed or fraction of a photographed object's size. Keeping its meaning attached to the route makes the calculation usable again for the other points without introducing a physical assumption.

Derive the remaining three heights

For B = (6, 4), a point on the ray has coordinates (6t, 4t). Setting its horizontal coordinate to two gives t = 1/3. Its height on the image line is then four-thirds. Put B's mark at (2, 4/3), below A's mark.

For C = (8, 4), the image line is one-quarter of the horizontal distance from O to C. The corresponding height is one-quarter of four, or one. C's mark is (2, 1). The object's height has remained four; only the fraction of its ray reached at the image line has changed.

For D = (10, 4), solve 10t = 2. The fraction is one-fifth, giving image height four-fifths. D's mark is (2, 4/5). Keep that exact fraction in the working even if you use 0.8 as an additional plotting note.

The four image heights, from top to bottom, are two, four-thirds, one and four-fifths. Their order follows these calculations. Each mark retains the identity of the object point that produced it, so you can inspect one ray without having to infer its endpoint from the finished collection of dots.

Compare the gaps after plotting the marks

Subtract neighboring image heights. From A's mark down to B's, the gap is 2 minus 4/3, or 2/3. From B's down to C's, it is 4/3 minus 1, or 1/3. From C's down to D's, it is 1 minus 4/5, or 1/5.

These gaps are unequal even though all three object-depth intervals were two. The first projected gap is twice the second. The third is smaller again. Those comparisons concern this declared diagram; they do not report distances measured from a real photograph or establish how much detail a camera can resolve.

Add the three image gaps as a check. Two-thirds plus one-third is one; adding one-fifth gives six-fifths. The distance between the first and last image heights is also 2 minus 4/5, or six-fifths. This confirms the gap arithmetic, while each individual ray calculation still supplies its own mark.

Check an equally spaced substitute against the rays

Suppose a second drawing keeps the first image height at two and the last at four-fifths, then divides that span into three equal gaps. Each gap would be six-fifths divided by three, or two-fifths. Its four proposed heights would be 2, 8/5, 6/5 and 4/5.

That list preserves the endpoints and contains four marks. It still fails the projection rule for the two interior object points. B's ray reaches x = 2 at height 4/3, not 8/5. C's ray reaches it at height one, not 6/5. Matching the end marks does not relocate those ray intersections.

Repair the substitute by returning each interior mark to its own calculated height. There is no need to move A, D or the image line to accommodate the equal-gap list. If you deliberately choose equal image gaps instead, retain that as another drawing requirement with a different relationship to the original object points.

Keep the rule visible when changing the diagram

For any one of our object depths d, the point is (d, 4). Its ray has coordinates (td, 4t). At x = 2, the fraction is t = 2/d, so the projected height is 8/d. This short rule reproduces all four calculations without treating a list of rounded decimals as the original definition.

The gap between the projected marks for depths d and d + 2 can also be written directly. Subtract 8/(d + 2) from 8/d. Putting the two fractions over a common denominator gives 16 divided by d times (d + 2). The equal depth increment stays two; the denominator depends on where that increment begins.

For the first interval, d is four, so the denominator is four times six, or twenty-four. Sixteen twenty-fourths reduces to two-thirds. For the next interval it is six times eight, or forty-eight, giving one-third. For the last it is eight times ten, or eighty, giving one-fifth. This reproduces the three gaps without measuring their drawn appearance.

The calculation also identifies which inputs must stay together. A gap of one-third belongs to the interval from depth six to depth eight under the x = 2 image-line rule. It is not the common spacing of the complete object row or the complete image-mark row. Write both depth endpoints beside a gap when retaining only part of the diagram.

Now make a separate copy with the image line at x = 1. Keep O and the four object points unchanged. The fraction becomes 1/d, and the projected height becomes 4/d. The four new heights are 1, 2/3, 1/2 and 2/5. Each is half its counterpart in the first drawing.

The new gaps are 1/3, 1/6 and 1/10. They are also half the earlier gaps and remain unequal. Label this copy with x = 1. Its smaller heights follow from a changed image-line position; they are not corrections to the valid x = 2 calculations.

A hand sketch can illustrate the arrangement while its exact values stay in the accompanying note. A thick pencil mark or a casually placed dot cannot certify fractional accuracy. If a drawn mark seems inconsistent, check the written point, ray and image-line position before changing the mathematical input to match the sketch.

Read the two ruler accounts

The Bybiment handled ruler offer names a two-piece ruler set. Its sole image shows two graduated rulers with handles and integral bubble vials. These are offer statements and visible features; this tutorial supplies no accuracy, grip, corrosion or leveling result.

The HYKE flat ruler offer names ten rulers and 40cm (15 Inch) in its title, while its description says 40cm / 12 inches. Its sole image shows ten flat rulers marked to 40cm. The inconsistent inch figures remain unresolved; this drawing does not authenticate dimensions or delivered contents.

Leave neighboring drawing decisions with their own methods

The circle-and-compass drawing tutorial establishes a centre and radius before tracing an outline. Here, the image marks come from straight rays and their intersections with a fixed line. No compass adjustment or circle construction is needed for the stipulated model.

The paper plan of path length and occupied span separates distance along a connected route from its axis-aligned bounds. Our image spacing comes from central rays rather than dropping coordinates onto an axis or calculating the total length of those rays.

Keep a possible purchase outside the diagram

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Save the projected positions with their starting points

Retain O, the image line, the object row and the four named depths beside the final image heights. For x = 2, the marks belong at 2, 4/3, 1 and 4/5, with downward gaps 2/3, 1/3 and 1/5. Those values let another reader reconstruct the projection drawing spacing from the same inputs.

Keep the equal-gap substitute and the x = 1 copy labelled separately if you save them. One changes the proposed interior marks; the other changes the image-line position and recalculates every mark. The original result remains tied to its own viewpoint, straight rays and fixed image line.