Class: Ra::Shape::Sphere

Inherits:
Base
  • Object
show all
Defined in:
lib/ra/shape/sphere.rb

Overview

A sphere at origin <0,0,0> with a radius 1. A sphere surface is defined:

x² + y² + z² = radius²

A unit radius simplifies further:

x² + y² + z² = 1

A ray ‘x` / `y` / `z` values at `t` use the `origin` and `direction`:

x = origin.x + direction.x * t
y = origin.y + direction.y * t
z = origin.z + direction.z * t

Substituting ‘x` / `y` / `z` allows for solving for `t`:

1 = (origin.x + direction.x * t)²
  + (origin.y + direction.y * t)²
  + (origin.z + direction.z * t)²

Simplifying gives us a quadratic formula with terms defined as:

a = direction.x² + direction.y² + direction.z²
b = 2 * ((origin.x * direction.x) + (origin.y * direction.y) + (origin.z * direction.z))
c = origin.x² + origin.y² + origin.z² - 1
discriminant = b² - 4ac
t = (-b ± √discriminant) / (2a)

A discriminant <0 indicates the ray does not intersect the sphere.

Constant Summary collapse

RADIUS = 
1

Instance Attribute Summary

Attributes inherited from Base

#material

Instance Method Summary collapse

Methods inherited from Base

#color, #initialize, #intersect, #normal

Constructor Details

This class inherits a constructor from Ra::Shape::Base

Instance Method Details

#l_normal(point:) ⇒ Ra::Tuple

Parameters:

  • point (Vector)

Returns:



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# File 'lib/ra/shape/sphere.rb', line 69

def l_normal(point:)
  point - Vector[0, 0, 0, Ra::Tuple::VECTOR]
end

#t_intersect(ray:) ⇒ Array<Numeric>

Parameters:

Returns:

  • (Array<Numeric>) 


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# File 'lib/ra/shape/sphere.rb', line 56

def t_intersect(ray:)
  origin = ray.origin - Vector[0, 0, 0, Ra::Tuple::POINT]
  direction = ray.direction

  Quadratic.solve(
    a: direction.dot(direction),
    b: 2 * direction.dot(origin),
    c: origin.dot(origin) - RADIUS,
  )
end

#uv_point(point:) ⇒ Vector

Returns <u = 0.0..1.0, v = 0.0..1.0>.

Parameters:

  • point (Vector) —

    <x, y, z, Tuple::POINT>

Returns:

  • (Vector) —

    <u = 0.0..1.0, v = 0.0..1.0>



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# File 'lib/ra/shape/sphere.rb', line 39

def uv_point(point:)
  x = point[0]
  y = point[1]
  z = point[2]

  radius = Vector[x, y, z].magnitude
  theta = Math.atan2(x, z)
  phi = Math.acos(y / radius)

  u = 1 - ((theta / (2 * Math::PI)) + 0.5)
  v = 1 - (phi / Math::PI)

  Vector[u, v]
end