How to calculate correct plane-frustum intersection?

Question

Question:

I need to calculate intersection shape (purple) of plane defined by Ax + By + Cz + D = 0 and frustum defined by 4 rays emitting from corners of rectangle (red arrows). The result shoud be quadrilateral (4 points) and important requirement is that result shape must be in plane's local space. Plane is created with transformation matrix T (planes' normal is vec3(0, 0, 1) in T's space).

Explanation:

This is perspective form of my rectangle projection to another space (transformation / matrix / node). I am able to calculate intersection shape of any rectangle without perspective rays (all rays are parallel) by plane-line intersection algorithm (pseudocode):

Definitions:

// Plane defined by normal (A, B, C) and D
struct Plane { vec3 n; float d; };

// Line defined by 2 points
struct Line { vec3 a, b; };

Intersection:

vec3 PlaneLineIntersection(Plane plane, Line line) {
    vec3 ba = normalize(line.b, line.a);
    float dotA = dot(plane.n, l.a);
    float dotBA = dot(plane.n, ba);
    float t = (plane.d - dotA) / dotBA;
    return line.a + ba * t;
}

Perspective form comes with some problems, because some of rays could be parallel with plane (intersection point is in infinite) or final shape is self-intersecting. Its works in some cases, but it's not enough for arbitary transformation. How to get correct intersection part of plane wtih perspective?

Simply, I need to get visible part of arbitary plane by arbitary perspective "camera".

Thank you for suggestions.

Ripi2 Ripi2 · Accepted Answer · 2017-09-20T00:08:47

Intersection between a plane (one Ax+By+Cx+D equation) and a line (two planes equations) is a matter of solving the 3x3 matrix for x,y,z.

Doing all calculations on T-space (origin is at the top of the pyramid) is easier as some A,B,C are 0.

What I don't know if you are aware of is that perspective is a kind of projection that distorts the z ("depth", far from the origin). So if the plane that contains the rectangle is not perpendicular to the axis of the fustrum (z-axis) then it's not a rectangle when projected into the plane, but a trapezoid.

Anyhow, using the projection perspective matrix you can get projected coordinates for the four rectangle corners.

To tell if a point is in one side of a plane or in the other just put the point coordinates in the plane equation and get the sign, as shown here

How to calculate correct plane-frustum intersection?

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Explanation:

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