Basic 3D

Perspective Divide

Divide x and y by z to show exact perspective foreshortening.

perspective-divide Perspective projection makes farther points smaller by dividing their x and y coordinates by z.

Example

Divide x and y by z to show exact foreshortening.

highlighted = computed this step

Step 1 — z equals 1

Compute the highlighted value.

point, factor, projection((4, 2, 1), 1, (4, 2))\begin{array}{c|c}\text{point, factor, projection}&\hlmath{\text{((4, 2, 1), 1, (4, 2))}}\end{array}
Perspective divideLarger z values divide x and y toward the origin.z1z2z4

Step 2 — z equals 2

Compute the highlighted value.

point, factor, projection((4, 2, 2), 1/2, (2, 1))\begin{array}{c|c}\text{point, factor, projection}&\hlmath{\text{((4, 2, 2), 1/2, (2, 1))}}\end{array}
Perspective divideLarger z values divide x and y toward the origin.z1z2z4

Step 3 — z equals 4

Compute the highlighted value.

point, factor, projection((4, 2, 4), 1/4, (1, 1/2))\begin{array}{c|c}\text{point, factor, projection}&\hlmath{\text{((4, 2, 4), 1/4, (1, 1/2))}}\end{array}
Perspective divideLarger z values divide x and y toward the origin.z1z2z4

Step 4 — Scale factors

Compute the highlighted value.

scale factors(1, 1/2, 1/4)\begin{array}{c|c}\text{scale factors}&\hlmath{\text{(1, 1/2, 1/4)}}\end{array}
Perspective divideLarger z values divide x and y toward the origin.z1z2z4