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5 days ago
2
A Ray is defined as:
\(\enclose{circle}[mathcolor="white"]{\color{white}{1}} \;\;\) \(P = P_0 + t * D\)
A Plane is defined as:
\(\enclose{circle}[mathcolor="white"]{\color{white}{2}} \;\;\) \(P \cdot N - C = 0\)
This formula is known as Hesse Normal Form. The symbols are interpreted as follows:
| \(P_0\) | ray origin |
| \(D\) | ray direction (unit vector) |
| \(N\) | plane normal (unit vector) |
| \(C\) | plane distance from axis origin |
Substituting \(P\) in \(\enclose{circle}[mathcolor="white"]{\color{white}{2}}\) from \(\enclose{circle}[mathcolor="white"]{\color{white}{1}}\):
\[(P_0 + t * D) \cdot N - C = 0\] \[(P_0 \cdot N) + t * (D \cdot N) - C = 0\] \[P_0 \cdot N - C = -t * (D \cdot N)\]
Solving for \(t\):
\(\enclose{circle}[mathcolor="white"]{\color{white}{3}} \;\;\) \(t = -\frac{(P_0 \cdot N - C)}{D \cdot N}\)
The ray is parallel to the plane and no intersection point exists if:
\[(D \cdot N) = 0\]
If \(t\) is negative it means that the ray intersects the plane behind the ray’s origin point, at the opposite direction. Otherwise, there is a single intersection point at position \(P\) with:
\[P = P_0 + t * D\]
Note the formula to calculate \(t\) can be manipulated a bit further to optimize the implementation.
Representing the plane distance as a dot product of a point \(X\) with the plane normal:
\[C = X \cdot N\]
Therefore:
\[-(P_0 \cdot N - C) = C - P_0 \cdot N = X \cdot N - P_0 \cdot N = (X - P_0) \cdot N\]
Substituting in \(\enclose{circle}[mathcolor="white"]{\color{white}{3}}\):
\[t = \frac{(X - P_0) \cdot N}{D \cdot N}\]
