# Essential ray tracing algorithms

## 2.Ray/sphere intersection and mapping

### 2.1 Intersection of the sphere – Algebraic Solution

calculate the intersection of the sphere and one ray from algebraic perspective

### 2.2 Intersection of the sphere – Geometric Solution

Use some geometric properties to deduce the computation amount.

### 2.3 Comparison of algebraic and geometric solution

Algebraic solution uses many replicate calculation like add first and subtract later while geometric solutions reduces this procedure. They in fact are internal unity.

I think there are others approaches in algebraic and geometric to speed up the ray tracing.

### 2.4 Precision problems

four methods are introduced in float-point calculations when solving intersections.

### 2.5 Spherical inverse mapping

map the sphere into one plane

## 3.Ray/plane algorithms

### 3.1 Ray/plane intersection

Given a plane normal, origin and director of one ray, find the intersection on the plane.
Note: Two-sided plane and one-sided plane are different when finding the intersection.

### 3.2 Polygon Intersection

Finding if one point on a plane is inside a polygon in that plane.

• Jordan Curve Theorem: two types introduced in the book, it determines whether the intersection is inside the polygon.

### 3.3 Convex quadrilateral Inverse mapping

Obtain the location of a point within the convex quadrilateral. Just give the conclusion no derivation.

Triangle inverse mapping is a special case.

## 4.Ray/Box intersection

Tell you how to determine if the ray intersects the box but don’t tell you why.

Give quadric equation to solve the intersection.
A guide to efficiency concern and floating point arithmetic imprecision.

### 5.2 Standard Inverse Mapping.

• circle
• cylinder
• cone
• others do not have std inverse mapping