Seeing Through a Projection

From a cube to its image

Begin with three parts of this demonstration: a point source above, a cube below it, and a receiving plane underneath. Rays from the source pass through the cube’s vertices and meet the plane. Those meeting points form an image of the cube. This is a central projection from three dimensions onto two.

Come close to the source cube. It has eight vertices and twelve edges, just as before. Each vertex has a corresponding point on the receiving plane. We will keep those identities throughout the demonstration, even when the image changes shape. Projection gives us another representation of the cube, rather than another cube with different connections.

Follow one vertex down its ray toward the receiving plane. The original vertex and its projected point lie on the same straight line from the source. The line determines where the image goes. Repeating that construction for all eight vertices gives the projected endpoints of all twelve edges.

Now focus on the image itself. The source cube is temporarily hidden so the receiving plane is easier to read. All eight projected points lie on this flat plane. The edges join the same corresponding endpoints as before, but their apparent lengths and directions differ from the edges of the original cube.

Move the source, keep the cube

Watch the light source move nearer to the cube. The cube stays fixed, and the rays now spread more strongly before reaching the plane. The image grows, but not by one common factor for every part. Vertices at different distances from the source receive different amounts of enlargement.

Inspect the resulting image. With this unrotated cube, the upper square face is closer to the source and makes the larger square in the projection. The lower face makes the smaller square. The original faces are congruent. Their different projected sizes come from their different positions along the rays.

Move the source farther away again, beyond its starting distance. The cube still has the same shape and position. The rays become more nearly parallel across the cube, and the difference in enlargement between its upper and lower faces decreases. Follow that change in the image while the source recedes.

Look at the image from this farther source. The two face images are now closer in size. They are not exactly equal, because the source is still at a finite distance. Parallel projection is the limiting comparison as that distance grows without bound. The control here approaches that situation; it does not place the source at infinity.

A changing image, the same connections

Keep the source at its current position and rotate the cube. The source and receiving plane stay fixed while the original vertices move. Each projected point follows its corresponding ray, so the image changes continuously. A changing outline does not mean the cube has gained or lost vertices, edges or faces.

Approach the cube as it turns a little farther. Its edges still connect the same pairs of vertices, and its square faces remain square in the original three-dimensional geometry. The projected picture can look quite different from a familiar cube drawing. Its appearance depends on orientation as well as the position of the source.

Pause the rotation at this view and come close to a crossing in the image. These two projected edges cross, but there is no vertex marker at their intersection. They represent edges separated in the original cube. A crossing in a flat picture does not by itself create a new mathematical connection.

Pull back to the cube and its image together. The original still has eight vertices and twelve edges. Use those indexed connections to read the projection, rather than treating every visible crossing as a corner. The picture carries the connections we projected, but it also introduces overlaps that belong only to the picture.

What the plane cannot retain

Watch a translucent copy travel from the cube toward its image. Each vertex follows the same ray that determined its projected point. This moving copy is a visual aid for the correspondence. The original cube stays in place, and the receiving plane stays fixed while the copy approaches the image.

At the endpoint, the copy coincides with the image, so only the projected layer remains visible here. Every projected vertex now lies on one plane. The original vertices had different heights above it. Those heights cannot remain as an independent direction within this two-dimensional receiving surface.

Follow another corresponding vertex in an intermediate copy. A point moving along this ray keeps the same intersection with the receiving plane. Several positions in space can therefore lead to the same projected point. The image records where the ray meets the plane, without recording a unique distance along that ray.

Return to the complete image. Because we already know the source object is a cube, its familiar connections help us interpret it. An image by itself does not uniquely specify every possible three-dimensional object that could produce it. Keep the original geometry, the projection rule and the resulting picture distinct as you read the demonstration.

One dimension higher

Now return to the tesseract. This display uses a projection from four dimensions into three, followed by the ordinary screen view. The nested cubes are a way to read its connections, not evidence that one mathematical cube is physically smaller than another. Recall the two congruent cube copies from the construction lesson.

Come close to the cube copy that appears smaller. Its eight vertices and twelve edges are still connected as a cube. The size difference in the image plays a role similar to the unequal square images in our earlier cube projection. The dimensional setting has changed, but correspondence still matters more than apparent size.

Reduce the perspective strength and watch the two cube images become closer in size. We are changing the projection, not stretching the tesseract. Its sixteen vertices and thirty-two edges keep their identities. The fourth coordinate influences the three-dimensional image, even though it is no longer displayed as an independent fourth direction.

Restore the stronger perspective. The nested arrangement becomes easier to separate again. This is a mathematical projection analogy; an ordinary lamp does not let us illuminate a four-dimensional object in our room. The calculation carries corresponding points into fewer dimensions, and the World lets us inspect that resulting representation.

Keep the original and compare the projection

Return to the unrotated cube with the source nearby. Keep the source cube and receiving plane in the same positions while comparing the image. A vertex on the face nearer the source casts a more enlarged image than a corresponding vertex on the farther face. The two original faces remain congruent. Their unequal images therefore cannot be used as direct measurements of their original side lengths.

Follow this selected vertex and its projected partner along their ray. Now let the translucent copy travel halfway toward the receiving plane. Its selected point changes position in space, but remains on the same ray from the source. Continuing that ray still reaches the same marked point on the plane. The halfway position is a correspondence aid, not an additional cube with preserved lengths.

Continue the copy to the plane. Its selected point arrives at the projected endpoint, and the copied vertices lose their separate heights. This is a concrete example of information that the image alone cannot supply: it identifies the ray’s intersection, but not a unique original location along that ray. Knowing the source object was a cube provides additional information beyond the image.

Restore the original cube and move the source farther away. The source moves beyond this close view, which keeps the cube and its receiving-plane image readable. Once the overview settles, its comparison scale stays fixed. Keep comparing the same two original faces as their projected sizes approach one another. The image changes while the source cube does not.

Follow another vertex at this farther source position. Its ray still determines its projected partner by the same rule. The rays are more nearly parallel across the cube, but the source is at a finite distance, so the enlargement is not exactly equal for all depths. A useful limiting idea should not erase the remaining difference in the example actually displayed.

Separate the three kinds of movement

Return to the cube demonstration and move the camera around it. The cube, the source and the projected points stay fixed in the World. Only our viewing position changes. Compare this with the earlier cube rotation, where the original vertices moved and their projected points moved with them.

Now hold that camera direction while the source moves nearer. The cube remains fixed, but the actual projected points move across the receiving plane. That is a change in the projection itself. Camera motion, object rotation and source motion can all change a screen picture, but they are different operations.

Inspect one projected point and the original vertex connected to it. You can check the correspondence directly along their ray, whatever the current camera angle. This relationship is the stable reference through the changing pictures. If the image becomes confusing, return to a vertex, its connecting ray and its original place in the cube.

Finish with the tesseract projection. Its apparent sizes and overlaps depend on representation, while its indexed connections describe the original object. In the next lesson we will rotate through a fourth direction. Keep today’s distinction ready: changing the higher-dimensional object’s orientation is different from moving our camera around its projected image.