Dr. Sergey Kosov
University Lecturer and Entrepreneur
Contents
- Ray-Sphere Intersection
- Ray-Plane Intersection
- Ray-Triangle Intersection
- Ray-Box Intersection
Lesson 3:
Ray-Geometry Intersection Algorithms
Basic Math
Ray Parametrization
\vec{p}(t)=\vec{o}+t\vec{d}
Origin point:
\vec{o}\in\mathbb{R}^3
Direction unit vector:
\vec{d}\in\mathbb{R}^3
Distance from origin:
t\in\mathbb{R}
Sphere Parametrization
Point on the ray:
\vec{p}\in\mathbb{R}^3
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
Origin point:
\vec{c}\in\mathbb{R}^3
Sphere radius:
r\in\mathbb{R}
Point on the sphere:
\vec{p}\in\mathbb{R}^3
\vec{c}
\vec{o}
t=1
t=2
t=3
\vec{d}
\vec{p}_1
\vec{p}_2
\vec{p}_1-\vec{c}
\vec{p}_2-\vec{c}
Basic Math
Ray Parametrization
Sphere Parametrization
\vec{o}
t=1
t=2
t=3
\vec{p}_1-\vec{c}
\vec{p}_2
\vec{p}_2-\vec{c}
struct Ray {
Vec3f org; // Origin
Vec3f dir; // Direction
double t; // Current/maximum hit distance
} ray;class rt::CPrimSphere
{
Vec3f m_origin; // Position of the sphere's center
float m_radius; // Radius of the sphere
};\vec{c}
\vec{p}_1
\vec{d}
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
Ray-Sphere Intersection
Ray Parametrization
Sphere Parametrization
\vec{o}
\vec{d}
\vec{c}
\vec{L}=\vec{c}-\vec{o}
Geometrical Approach
\vec{x}
\vec{b}
\vec{b}=\vec{o}+t_b\vec{d}
\vec{x}=\vec{o}+t_x\vec{d}
t_b = \vec{d}\cdot\vec{L}
\Delta
t_x=t_b-\Delta
r
h
h^2=|\vec{L}|^2-t^{2}_{b}
h^2=r^2-\Delta^2
|\vec{L}|^2-t^{2}_{b}=r^2-\Delta^2
\Delta = \sqrt{r^2-|\vec{L}|^2+t^{2}_{b}}
Find closest intersection point
Introduce auxiliary point
Find
\vec{x}
\vec{b}
\Delta
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
Ray-Sphere Intersection
Ray Parametrization
Sphere Parametrization
\vec{o}
\vec{d}
\vec{L}=\vec{c}-\vec{o}
\vec{c}
Please also mind special cases:
1. Only one intersection point
h=r
\vec{x}=\vec{o}+t_x\vec{d}
t_x=t_b
Find closest intersection point
\vec{b}
h=r
Geometrical Approach
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
\Delta = 0
Ray-Sphere Intersection
Ray Parametrization
Sphere Parametrization
\vec{o}
\vec{d}
\vec{L}=\vec{c}-\vec{o}
\vec{c}
h>r
r^2-|\vec{L}|^2+t^{2}_{b}<0
Please also mind special cases:
2. No intersection points
\vec{x}=\vec{o}+t_x\vec{d}
t_x=t_b-\Delta
\Delta =undefined
Find closest intersection point
\vec{b}
Geometrical Approach
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
h>r
Ray-Sphere Intersection
Ray Parametrization
Sphere Parametrization
Please also mind special cases:
3. Ray origin lies in the sphere
\vec{x}=\vec{o}+t_x\vec{d}
t_x=t_b+\Delta
\Delta = \sqrt{r^2-|\vec{L}|^2+t^{2}_{b}}
Find closest intersection point
Geometrical Approach
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
\vec{o}
\vec{d}
\vec{L}
\vec{c}
h
\vec{b}
\Delta
|\vec{L}| < r
Ray-Sphere Intersection
Ray Parametrization
Sphere Parametrization
Substitute ray equation into the sphere equation and solve it for
(\vec{o}+t\vec{d}-\vec{c})\cdot(\vec{o}+t\vec{d}-\vec{c})-r^2=0
t
(t\vec{d} + (\vec{o}-\vec{c}))^2-r^2=0
t^2|\vec{d}|^2 - 2t\vec{d}\cdot\vec{L} + |\vec{L}|^2-r^2=0
Solve it as a quadratic equation
t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
1t^2 + (-2\vec{d}\cdot\vec{L})t + (|\vec{L}|^2-r^2)=0
(t\vec{d} - \vec{L})^2-r^2=0
b
c
t_{1,2}=\vec{d}\cdot\vec{L}\pm\sqrt{r^2-|\vec{L}|^2+(\vec{d}\cdot\vec{L})^2}
a
Analytical Approach
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{c})\cdot(\vec{p}-\vec{c})=r^2
Ray-Plane Intersection
Ray Parametrization
Plane Parametrization
\vec{p}(t)=\vec{o}+t\vec{d}
(\vec{p}-\vec{a})\cdot\vec{n}=0
Point on the plane:
\vec{a}\in\mathbb{R}^3
Normal to the plane:
\vec{n}\in\mathbb{R}^3
Point on the plane:
\vec{p}\in\mathbb{R}^3
Substitute ray equation into the plane equation and solve it for
(\vec{o}+t\vec{d}-\vec{a})\cdot\vec{n}=0
t
t\vec{d}\cdot\vec{n} + (\vec{o}-\vec{a})\cdot\vec{n}=0
t\vec{d}\cdot\vec{n} = (\vec{a}-\vec{o})\cdot\vec{n}
Analytical Approach
t = \frac{(\vec{a}-\vec{o})\cdot\vec{n}}{\vec{d}\cdot\vec{n}}
There might be 0 or 1 or infinitely many solutions
\vec{n}
\vec{a}
\vec{p}
Ray-Triangle Intersection
Barycentric Coordinates
In order to represent point p lying on the segment defined by known points b and c we can use the same logic which lies behind the ray definition:
\vec{c}
\vec{q}
\vec{q} = \vec{b} + t(\vec{c} - \vec{b})
\vec{q} = \vec{b} + t\vec{c} - t\vec{b}
\vec{q} = (1-t)\vec{b} + t\vec{c}
\vec{b}
Ray-Triangle Intersection
Barycentric Coordinates
By analogy to represent point q lying on the segment defined by known points a and q we can write:
\vec{c}
\vec{q}
\vec{q} = \vec{b} + t(\vec{c} - \vec{b})
\vec{q} = \vec{b} + t\vec{c} - t\vec{b}
\vec{q} = (1-t)\vec{b} + t\vec{c}
\vec{b}
\vec{a}
\vec{p}
\vec{p} = \vec{q} + s(\vec{a} - \vec{q})
\vec{p} = \vec{q} + s\vec{a} - s\vec{q}
\vec{p} = (1-s)\vec{q} + s\vec{a}
\vec{p} = (1-s)\left((1-t)\vec{b} + t\vec{c}\right) + s\vec{a}
\vec{p} = s\vec{a} + (1-s)(1-t)\vec{b} + (1-s)t\vec{c}
u
v
w
1 = s + (1-s)(1-t) + (1-s)t
Ray-Triangle Intersection
Barycentric Coordinates
Thus any point p within a triangle abc may be expressed as a linear combination of its vertices:
\vec{c}
\vec{q}
\vec{b}
\vec{a}
\vec{p}
\vec{p} = w\vec{a} + u\vec{b} + v\vec{c}
w+u+v=1
where parameters
or
\vec{p} = (1-u-v)\vec{a} + u\vec{b} + v\vec{c}
are called barycentric coordinates
Ray-Triangle Intersection
Barycentric Coordinates
Geometrical meaning
\vec{c}
\vec{b}
\vec{a}
\vec{p}
\vec{p} = (1-u-v)\vec{a} + u\vec{b} + v\vec{c}
u
v
1-u-v
u=\frac{S_{apc}}{S_{abc}}
v=\frac{S_{abp}}{S_{abc}}
1-u-v=\frac{S_{pbc}}{S_{abc}}
Where S designates the signed area of a triangle
Ray-Triangle Intersection
Möller-Trumbore Algorithm
Ray Parametrization
\vec{p}(t)=\vec{o}+t\vec{d}
Triangle Parametrization
\vec{p}(u,v)=(1-u-v)\vec{a}+u\vec{b}+v\vec{c}
Triangle vertices:
\vec{a}, \vec{b}, \vec{c}\in\mathbb{R}^3
u,v\in\mathbb{R}
Barycentric coordinates:
\vec{a}
\vec{b}
\vec{c}
Substitute ray equation into the triangle equation and solve it for
\vec{o}+t\vec{d}=(1-u-v)\vec{a}+u\vec{b}+v\vec{c}=0
Analytical Approach
t
\begin{bmatrix}
-\vec{d} &
\vec{b}-\vec{a} &
\vec{c}-\vec{a}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{o}-\vec{a}
\vec{e_1}
\vec{e_2}
\begin{bmatrix}
-\vec{d} &
\vec{e_1} &
\vec{e_2}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{T}
Point on the triangle:
\vec{p}\in\mathbb{R}^3
\vec{p}(u,v)
\vec{o}
\vec{T}
t
Ray-Triangle Intersection
Möller-Trumbore Algorithm
\begin{bmatrix}
-\vec{d} &
\vec{e_1} &
\vec{e_2}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{T}
Using the Cramer’s Rules for Systems of Linear Equations with Three Variables:
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \frac{1}{
\begin{vmatrix}
-\vec{d} & \vec{e_1} & \vec{e_2}
\end{vmatrix}
}
\begin{pmatrix}
\begin{vmatrix}
\vec{T} & \vec{e_1} & \vec{e_2}
\end{vmatrix} \\
\begin{vmatrix}
-\vec{d} & \vec{T} & \vec{e_2}
\end{vmatrix} \\
\begin{vmatrix}
-\vec{d} & \vec{e_1} & \vec{T}
\end{vmatrix}
\end{pmatrix}
\begin{vmatrix}
-\vec{d} & \vec{e_1} & \vec{e_2}
\end{vmatrix}
= \begin{vmatrix}
-d_x & e_{1x} & e_{2x} \\
-d_y & e_{1y} & e_{2y} \\
-d_z & e_{1z} & e_{2z}
\end{vmatrix}
= \begin{vmatrix}
-d_x & -d_y & -d_z \\
e_{1x} & e_{1y} & e_{1z} \\
e_{2x} & e_{2y} & e_{2z}
\end{vmatrix}
=-\vec{d}\cdot\left(\vec{e_1}\times\vec{e_2}\right)
=\vec{d}\cdot\left(\vec{e_2}\times\vec{e_1}\right)
=
=\left(\vec{d}\times\vec{e_2}\right)\cdot\vec{e_1}
determinant:
Ray-Triangle Intersection
Möller-Trumbore Algorithm
\begin{bmatrix}
-\vec{d} &
\vec{e_1} &
\vec{e_2}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{T}
Using the Cramer’s Rules for Systems of Linear Equations with Three Variables:
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \frac{1}{
\begin{vmatrix}
-\vec{d} & \vec{e_1} & \vec{e_2}
\end{vmatrix}
}
\begin{pmatrix}
\begin{vmatrix}
\vec{T} & \vec{e_1} & \vec{e_2}
\end{vmatrix} \\
\begin{vmatrix}
-\vec{d} & \vec{T} & \vec{e_2}
\end{vmatrix} \\
\begin{vmatrix}
-\vec{d} & \vec{e_1} & \vec{T}
\end{vmatrix}
\end{pmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \frac{1}{
\left(\vec{d}\times\vec{e_2}\right)\cdot\vec{e_1}
}
\begin{pmatrix}
(\vec{T}\times\vec{e_1})\cdot\vec{e_2} \\
(\vec{d}\times\vec{e_2})\cdot\vec{T} \\
(\vec{T}\times\vec{e_1})\cdot\vec{d}
\end{pmatrix}
= \frac{1}{
\vec{P}\cdot\vec{e_1}
}
\begin{pmatrix}
\vec{Q}\cdot\vec{e_2} \\
\vec{P}\cdot\vec{T} \\
\vec{Q}\cdot\vec{d}
\end{pmatrix}
Ray-Triangle Intersection
Möller-Trumbore Algorithm
bool rt::CPrimTriangle::intersect(Ray& ray) const
{
const Vec3f pvec = ray.dir.cross(m_edge2);
const float det = m_edge1.dot(pvec);
if (fabs(det) < Epsilon)
return false;
const float inv_det = 1.0f / det;
const Vec3f tvec = ray.org - m_a;
float u = tvec.dot(pvec);
u *= inv_det;
if (u < 0.0f || u > 1.0f)
return false;
const Vec3f qvec = tvec.cross(m_edge1);
float v = ray.dir.dot(qvec);
v *= inv_det;
if (v < 0.0f || v + u > 1.0f)
return false;
float t = m_edge2.dot(qvec);
t *= inv_det;
if (ray.t <= t || t < Epsilon)
return false;
ray.t = t;
return true;
}\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \frac{1}{
\vec{P}\cdot\vec{e_1}
}
\begin{pmatrix}
\vec{Q}\cdot\vec{e_2} \\
\vec{P}\cdot\vec{T} \\
\vec{Q}\cdot\vec{d}
\end{pmatrix}
\vec{P} = \vec{d}\times\vec{e2}
\vec{T} = \vec{o}-\vec{a}
\vec{Q} = \vec{T}\times\vec{e1}
where
Ray-Triangle Intersection
Möller-Trumbore Algorithm
Geometrical Interpretation
\vec{a}
\vec{b}
\vec{c}
\vec{o}
\vec{d}
\begin{bmatrix}
-\vec{d} &
\vec{b}-\vec{a} &
\vec{c}-\vec{a}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{o}-\vec{a}
Ray-Triangle Intersection
Möller-Trumbore Algorithm
Geometrical Interpretation
\vec{b}-\vec{a}
\vec{c}-\vec{a}
\vec{o}-\vec{a}
\vec{d}
\begin{bmatrix}
-\vec{d} &
\vec{b}-\vec{a} &
\vec{c}-\vec{a}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{o}-\vec{a}
Translation
\vec{T}=\vec{o}-\vec{a}
Ray-Triangle Intersection
Möller-Trumbore Algorithm
Geometrical Interpretation
1
1
M^{-1}(\vec{o}-\vec{a})
\vec{d}
\begin{bmatrix}
-\vec{d} &
\vec{b}-\vec{a} &
\vec{c}-\vec{a}
\end{bmatrix}
\begin{pmatrix}
t \\
u \\
v
\end{pmatrix}
= \vec{o}-\vec{a}
Translation
\vec{T}=\vec{o}-\vec{a}
Change of base of the ray origin
M=
\begin{bmatrix}
-\vec{d} &
\vec{b}-\vec{a} &
\vec{c}-\vec{a}
\end{bmatrix}
u
v
Ray-Box Intersection
Ray Parametrization
Axis-Aligned Bounding Box (aabb)
\vec{p}(t)=\vec{o}+t\vec{d}
\vec{p}_{min},\vec{p}_{max}\in\mathbb{R}^3
Points bounding the box volume:
\vec{p}_{min}
\vec{p}_{max}
Bounded Volume
Ray-Box Intersection
A box is the intersection of 3 pairs of slabs in 3 dimensions
- Each slab contains two parallel planes and the space between them
We can detect the intersection between the ray and the box by detecting the intersections between the ray and the three pairs of slabs
- For each pair of slabs there are two intersection points tnear and tfar, and there are 6 intersection points in total for 3 pairs
- If the maximal tnear is ahead of the minimal tfar on the ray, the ray misses the box. Otherwise it hits the box.
\vec{p}_{min}
Bounded Volume
\vec{p}_{min}
Bounded Volume
\vec{o}
t^{near}_{y}
t^{far}_{y}
t^{near}_{x}
t^{far}_{x}
t^{near}_{y}
t^{far}_{y}
t^{near}_{x}
t^{far}_{x}
\vec{o}
\max_{i\in\{x,y,z\}}t^{near}_{i} < \min_{i\in\{x,y,z\}}t^{far}_{i}
Ray-Box Intersection
\vec{p}_{min}
Bounded Volume
\vec{p}_{min}
Bounded Volume
\vec{o}
t^{near}_{y}
t^{far}_{y}
t^{near}_{x}
t^{far}_{x}
t^{near}_{y}
t^{far}_{y}
t^{near}_{x}
t^{far}_{x}
\vec{o}
\max_{i\in\{x,y,z\}}t^{near}_{i} < \min_{i\in\{x,y,z\}}t^{far}_{i}
bool intersect(Ray& ray) {
calculate tnear.x , tfar.x , tnear.y , tfar.y , tnear.z , tfar.z on 3 axes;
tnear = max(tnear.x , tnear.y , tnear.z);
tfar = min(tfar.x , tfar.y , tfar.z);
if (tnear < tfar && tnear < ray.t) {
ray.t = tnear; // report intersection at tnear
return true;
}
return false; // no intersection
}
Ray-Geometry Intersection Algorithms
By Dr. Sergey Kosov
