Self-Assessment Solutions

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: c ² = a ² + b ² .

Distance Formula in 2D

where P ₁ ( x ₁ , y ₁ ) and P ₂ ( x ₂ , y ₂ ) are points on the line.

Distance Formula in 3D

where P ₁ ( x ₁ , y ₁ , z ₁ ) and P ₂ ( x ₂ , y ₂ , z ₂ ) are points on the line.

Midpoint Formula in 2D

is the midpoint between P ₁ ( x ₁ , y ₁ ) and P ₂ ( x ₂ , y ₂ ).

Midpoint Formula in 3D

is the midpoint between P ₁ ( x ₁ , y ₁ , z ₁ ) and P ₂ ( x ₂ , y ₂ , z ₂ ).

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 = r ²

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

Equation of a Circle Centered at the Origin

x ² + y ² = r ²

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

Equation of a Sphere

( x “ h ) ² + ( y “ k ) ² + ( z “ l ) ² = r ²

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

Equation of a Sphere Centered at the Origin

x ² + y ² + z ² = r ²

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 ( h ₂ “ h ₁ ) ² + ( k ₂ “ k ₁ ) ² ( r ₁ + r ₂ ) ² , a collision occurs.

Optimized Sphere-Sphere Collision Detection

Given two spheres and , if ( h ₂ “ h ₁ ) ² + ( k ₂ “ k ₁ ) ² + ( l ₂ “ i ₁ ) ² ( r ₁ + r ₂ ) ² , a collision occurs.