Vanishing Points

Recall: If we are drawing the perspective image of a line in the real-world, the vanishing point of our image is the point on the picture plane where the line of sight of the viewer is parallel to the real-world line -- this corresponds to the instant at which the viewer can no longer see the line.

The Vanishing Point Theorem
Suppose we have 2 (or more) lines in the real world that are parallel to each other, but not to the picture plane. Then their perspective images will not be parallel, but will all intersect at the same point. The point where they intersect is their vanishing point.

	if our lines are orthogonal to the picture plane (parallel to the z-axis), then the vanishing point will be directly opposite the viewer's eye, at the origin. The vanishing point of all the orthogonals is called the primary vanishing point.
We call the horizontal line through the primary vanishing point the horizon line. Since the primary vanishing point is at the origin, the horizon line corresponds to the x-axis.
	if our lines are still parallel to the "floor" (the xz-plane) but no longer orthogonal to the picture plane, then their vanishing point will no longer be at the origin, but will still be on the horizon line/x-axis. This is because if the viewer's line of sight is parallel to the line we're looking at, the it's parallel to the xz-plane. Since the line of sight begins in the xz-plane (the viewer's eye is on the z-axis), it must stay in the xz-plane. Thus it must hit the picture plane on the x-axis somewhere. (If the lines are parallel to the "side wall" (the yz-plane) then the vanishing point will be on the y-axis, which we'll call the verizon line.)
	There is no reason why any of the vanishing points needs to be on the canvas.

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