Chapter 2, "Geometry Snippets"

Pythagorean Theorem

In a right triangle, the square of the length c of the hypotenuse is equal to the sum of the squares of the lengths a and b of the other two sides: c² = a² + b².

Distance Formula in 2D

where P₁(x₁,y₁) and P₂(x₂,y₂) are points on the line.

Distance Formula in 3D

where P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂) are points on the line.

Midpoint Formula in 2D

is the midpoint between P₁(x₁,y₁) and P₂(x₂,y₂).

Midpoint Formula in 3D

is the midpoint between P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂).

Parabola with a Vertical Axis

y = a(x – h)² + k, with vertex (h,k) and axis of symmetry x = h.

Parabola with a Horizontal Axis

x = a(y – k)² + h, with vertex (h,k) and axis of symmetry y = k.

Equation of a Circle

(x – h)² + (y – k)² = r²

where the center is (h,k) and the radius is r.

Equation of a Circle Centered at the Origin

x² + y² = r²

where the center is (0,0) and the radius is r.

Equation of a Sphere

(x – h)² + (y – k)² + (z – l)² = r²

where the center is (h,k,l) and the radius is r.

Equation of a Sphere Centered at the Origin

x² + y² + z² = r²

where the center is (0,0,0) and the radius is r.

Circle-Circle Collision Detection

Given two circles and , if , a collision occurs.

Optimized Circle-Circle Collision Detection

Given two circles and , if (h₂ – h₁)² + (k₂ – k₁)² (r₁ + r₂)², a collision occurs.

Optimized Sphere-Sphere Collision Detection

Given two spheres and , if (h₂ – h₁)² + (k₂ – k₁)² + (l₂ – i₁)² (r₁ + r₂)², a collision occurs.