Great question! Checking all four rectangle vertices works, but it’s not the most efficient approach—especially when dealing with large numbers of rectangles or triangles. Let’s break down a faster method that combines early rejection and half-plane inclusion checks to minimize unnecessary computations.

Step 1: Early Rejection with Axis-Aligned Bounding Boxes (AABB) #

First, we can quickly eliminate any rectangles that can’t possibly fit inside the triangle by comparing their bounding boxes:

• For the triangle, compute its AABB by finding the minimum/maximum x and y values across its three vertices: (tri_min_x, tri_min_y) and (tri_max_x, tri_max_y)
• For the axis-aligned rectangle, its AABB is trivial to get (just use its min/max x/y from its corners)

If the two AABBs don’t overlap (e.g., rect_max_x < tri_min_x, or rect_min_y > tri_max_y, etc.), we can immediately return false—no need for further checks. This is an O(1) operation and filters out most invalid cases right away.

Step 2: Half-Plane Inclusion Checks #

A triangle is essentially the intersection of three half-planes (each side of the triangle defines a half-plane, and the triangle is the area that’s on the "inside" side of all three). For an axis-aligned rectangle to be fully inside the triangle, it must lie entirely within all three of these half-planes.

How to Define the Half-Planes #

For each edge of the triangle (formed by vertices P1 and P2):

1. Calculate the edge’s direction vector: dx = P2.x - P1.x, dy = P2.y - P1.y
2. Generate a normal vector pointing toward the triangle’s interior. Start with the counter-clockwise rotated normal: a = -dy, b = dx
3. Compute the line equation for the edge: a*x + b*y + c = 0, where c = dy*P1.x - dx*P1.y
4. Check if the triangle’s third vertex P3 satisfies a*P3.x + b*P3.y + c ≤ 0. If not, flip the signs of a, b, and c to ensure the inequality represents the interior half-plane.

Optimized Check for Axis-Aligned Rectangles #

Instead of checking all four rectangle vertices against each half-plane, we only need to check the extreme point of the rectangle relative to the half-plane’s direction:

• For the inequality a*x + b*y + c ≤ 0, find the rectangle’s point that maximizes a*x + b*y:
  • If a > 0, use the rectangle’s maximum x value; if a < 0, use the minimum x
  • If b > 0, use the rectangle’s maximum y value; if b < 0, use the minimum y
• Plug this extreme point into the inequality. If it satisfies the condition, the entire rectangle does (since a*x + b*y is linear, its maximum over the rectangle will be at this extreme point).

Run this check for all three half-planes. If all pass, the rectangle is fully inside the triangle; if any fail, it’s not.

Why This Is Faster #

• Early rejection skips expensive checks for obvious non-overlapping cases
• Each half-plane check only requires 1 calculation (instead of 4 for all vertices), cutting down total computations by 75% for this step
• We can exit early as soon as one half-plane check fails, avoiding unnecessary work

Example Pseudocode #

def is_rect_inside_triangle(rect_vertices, tri_vertices):
    # Step 1: Early rejection with AABB
    tri_min_x = min(p[0] for p in tri_vertices)
    tri_max_x = max(p[0] for p in tri_vertices)
    tri_min_y = min(p[1] for p in tri_vertices)
    tri_max_y = max(p[1] for p in tri_vertices)
    
    rect_min_x = min(p[0] for p in rect_vertices)
    rect_max_x = max(p[0] for p in rect_vertices)
    rect_min_y = min(p[1] for p in rect_vertices)
    rect_max_y = max(p[1] for p in rect_vertices)
    
    if (rect_max_x < tri_min_x or rect_min_x > tri_max_x or
        rect_max_y < tri_min_y or rect_min_y > tri_max_y):
        return False
    
    # Helper to get interior half-plane for a triangle edge
    def get_interior_halfplane(p1, p2, p3):
        dx = p2[0] - p1[0]
        dy = p2[1] - p1[1]
        a = -dy
        b = dx
        c = dy * p1[0] - dx * p1[1]
        
        # Flip normal if p3 is outside the initial half-plane
        if a * p3[0] + b * p3[1] + c > 0:
            a, b, c = -a, -b, -c
        return (a, b, c)
    
    # Get the three half-planes for the triangle
    hp1 = get_interior_halfplane(tri_vertices[0], tri_vertices[1], tri_vertices[2])
    hp2 = get_interior_halfplane(tri_vertices[1], tri_vertices[2], tri_vertices[0])
    hp3 = get_interior_halfplane(tri_vertices[2], tri_vertices[0], tri_vertices[1])
    
    # Check each half-plane
    for a, b, c in [hp1, hp2, hp3]:
        # Find the rectangle's extreme point for this half-plane
        x = rect_max_x if a > 0 else rect_min_x if a < 0 else rect_min_x
        y = rect_max_y if b > 0 else rect_min_y if b < 0 else rect_min_y
        
        if a * x + b * y + c > 0:
            return False
    
    return True

Extra Optimizations #

• Cache the triangle’s AABB and half-plane parameters if you’re checking multiple rectangles against the same triangle
• If a triangle edge is axis-aligned (e.g., horizontal), the half-plane check simplifies to a simple coordinate comparison (no need for the linear calculation)

内容的提问来源于stack exchange，提问作者Arutyun Enfendzhyan