3D AABBs

In what follows are various notes and equations dealing with 3D AABBs.

AABB of Points

• \(A\) is the bottom left of the AABB
• \(B\) is the top right of the AABB
• \((x_1, y_1, z_1), (x_2, y_2, z_2), \ldots, (x_n, y_n, z_n)\) are the points

• \(A_x = \min(x_1, x_2, \ldots, x_n)\)
• \(A_y = \min(y_1, y_2, \ldots, y_n)\)
• \(A_z = \min(z_1, z_2, \ldots, z_n)\)
• \(B_x = \max(x_1, x_2, \ldots, x_n)\)
• \(B_y = \max(y_1, y_2, \ldots, y_n)\)
• \(B_z = \max(z_1, z_2, \ldots, z_n)\)

Point-AABB Hit Detection

• \((x, y, z)\) is the point
• \(A\) is the bottom left of the AABB
• \(B\) is the top right of the AABB

No collision iff
\(x \lt A_x\) or \(x \gt B_x\) or
\(y \lt A_y\) or \(y \gt B_y\) or
\(z \lt A_z\) or \(z \gt B_z\)

AABB-AABB Hit Detection

• \(A\) is the bottom left of the first AABB
• \(B\) is the top right of the first AABB
• \(C\) is the bottom left of the second AABB
• \(D\) is the top right of the second AABB

No collision iff
\(B_x \lt C_x\) or \(A_x \gt D_x\) or
\(B_y \lt C_y\) or \(A_y \gt D_y\) or
\(B_z \lt C_z\) or \(A_z \gt D_z\)