已知三点求平面偏移量及三角形平面中P1点的求解方法

Alright, let's break this down step by step for both of your questions—this stuff is core to collision detection, so I get why it's tripping you up!

First, let's start with the standard plane equation: n · (P) + d = 0, where:

• n is the unit normal vector of the plane
• P is any point (x,y,z) lying on the plane
• d is the offset (or "plane constant") you're trying to find

Here's the step-by-step process to calculate d from three points A, B, C:

1. Compute two vectors that lie on the plane:
   • vecAB = B - A (subtract corresponding coordinates: e.g., B.x - A.x, B.y - A.y)
   • vecAC = C - A
2. Calculate the plane's normal using the cross product of these two vectors:
   • n = cross(vecAB, vecAC)
   • Normalize n to make it a unit vector (divide each component by its magnitude: |n| = sqrt(n.x² + n.y² + n.z²))
3. Compute the offset d using any of the three points (all will return the same result since they're on the plane):
   • d = -dot(n, A) (take the dot product of the unit normal and point A, then negate it)

Quick Pseudocode Example #

Vector3 vecAB = B - A;
Vector3 vecAC = C - A;
Vector3 n = cross(vecAB, vecAC).Normalize();
float d = -dot(n, A);

From your description, P1 is almost certainly the intersection point between your ray and the plane containing triangle ABC—this is the first critical step in ray-convex mesh collision detection (you then check if this point lies inside the triangle itself). Using A/B/C directly won't work because those are just triangle vertices, not the ray-plane intersection.

Here's how to compute P1 properly:

Step 1: Formalize your ray #

First, define your ray with a parametric equation:

• R(t) = R0 + t * Rd
  • R0: The starting point of your ray
  • Rd: The direction vector of your ray (unitize this first for consistent t values)
  • t: A scalar parameter (t ≥ 0 for points along the forward direction of the ray)

Step 2: Solve for t using the plane equation #

We already have the plane equation from Problem 1: n · P + d = 0. Substitute R(t) into this equation to solve for t:

n · (R0 + t*Rd) + d = 0
// Expand the dot product
dot(n, R0) + t*dot(n, Rd) + d = 0
// Rearrange to isolate t
t = -(dot(n, R0) + d) / dot(n, Rd)

Step 3: Calculate P1 (and handle edge cases) #

Once you solve for t, you need to validate it before computing P1:

• If abs(dot(n, Rd)) < 1e-6: The ray is parallel to the plane—there's no intersection (or the ray lies entirely on the plane, a rare edge case to handle separately)
• If t < 0: The intersection is behind the ray's starting point—ignore this for forward-facing collision checks
• If t is valid, compute P1:

  P1 = R0 + t * Rd
  

Step 4: Verify P1 is inside the triangle (critical for collision) #

Even if you have P1, you need to confirm it's inside triangle ABC to count as a collision. A reliable method uses cross products:

1. Compute vectors from A to P1 and A to B: vecAP = P1 - A, vecAB = B - A
2. Calculate cross1 = cross(vecAB, vecAP)—check if it aligns with the plane normal n (dot product should be ≥ 0)
3. Repeat for the other two edges:
   • vecBP = P1 - B, vecBC = C - B → cross2 = cross(vecBC, vecBP)
   • vecCP = P1 - C, vecCA = A - C → cross3 = cross(vecCA, vecCP)
4. If all three cross products have a non-negative dot product with n, P1 is inside the triangle.

Pseudocode for P1 Calculation #

// Assume we already have plane normal n (unit) and offset d from Problem 1
Vector3 R0 = ray.Start;
Vector3 Rd = ray.Direction.Normalize();

float denom = dot(n, Rd);
// Handle parallel ray/plane case
if (abs(denom) < 1e-6) {
    // No intersection or ray lies on plane
    return null;
}

float t = -(dot(n, R0) + d) / denom;
// Check if intersection is in front of the ray's start
if (t < 0) {
    return null;
}

Vector3 P1 = R0 + t * Rd;

// Now verify P1 is inside triangle ABC
// Implement the cross product check here...

内容的提问来源于stack exchange，提问作者René Jensen