An introduction to ray-tracing

Cast a ray from the viewer’s eye through each pixel on the screen to see where it hits some object view-plane object eye of viewer
Apply illumination principles from physics to determine the intensity of the light that is reflected from that spot toward the eye

• Mathematics problem: How can we find the spot where a ray hits a sphere?
• We apply our knowledge of vector-algebra
• Describe sphere’s surface by an equation
• Describe ray’s trajectory by an equation
• Then ‘solve’ this system of two equations
• There could be 0, 1, or 2 distinct solutions

• A sphere consists of points in space which lie at a fixed distance from a given center
• Let r denote this fixed distance (radius)
• Let c = ( c x
, c y
, c z
) be the sphere’s center
• If w = ( w x
, w y
, w z
) be any point in space, then distance( w , c ) is written ║ w – c ║
• Formula: ║ w – c ║= sqrt( ( w c )•( w c ) )
• Sphere is a set: { w ε R 3 | ( w c )•( w c ) = r }

• A ray is a geometrical half-line in space
• It has a beginning point: b = ( b x
, b y
, b z
)
• It has a direction-vector: d = ( d x
, d y
, d z
)
• It can be described with a parameter t ≥ 0
• If w = ( w x
, w y
, w z
) is any point in space, then w will be on the half-line just in case w = b + t d for some choice of the parameter t ≥ 0 .

• To find WHERE a ray hits a sphere, we think of w as a point traveling in space, and we ask: WHEN will w hit the shere?
• We can ‘substitute’ the formula for a point on the half-line into our formula for points that belong to the sphere’s surface, getting an equation that has t as its only variable
• It’s easy to ‘solve’ such an equation for t to find when w ‘hits’ the sphere’s surface

• Sphere:
• Half-line:
║ w – c ║ = r w = b + t d , t ≥ 0
• Substitution: ║ b + t d – c ║ = r
• Replacement: Let q = c – b
• Simplification: ║ t d – q ║ = r
• Square:
• Expand: t
(t
2 d d
– q
•d
)•(t
-2t d d –
•q q
+
) = r q
2
•q = r 2
• Transpose: t 2 d •d -2t d •q + ( q •q - r 2 ) = 0

• To solve: t 2 d •d -2t d •q + ( q •q - r 2 ) = 0
• Format: At 2 + Bt + C = 0
• Formula: t = -B/2A ± sqrt( B 2 – 4AC )/2A
• Notice: there could be 0, 1, or 2 hit times
• OK to use a simplifying assumption: d •d = 1
• Equation becomes: t 2 – 2t d • q + ( q • q – r 2 ) = 0
• Application: We want the ‘earliest’ hit-time: t = ( d • q ) - sqrt( ( d • q ) 2 – ( q • q – r 2 )
)

b
w2 w1 c half-line half-line
A half-line that begins inside the sphere will only ‘hit’ the spgere’s surface once

• Mathematical problem: How can we find the spot where a ray hits a plane?
• Again we can employ vector-algebra
• Any plane is determined by a two entities:
– a point (chosen arbitrarily) that lies in it, and
– a vector that is perpendicular (normal) to it
• Let p be the point and n be the vector
• Plane is the set: { w ε R 3 | ( w p )•n = 0 }

• Plane:
• Half-line:
( w – p ) • n = 0 w = b + t d , t ≥ 0
• Substitution: ( b + t d – p ) • n = 0
• Replacement: Let q = p – b
• Simplification: (t d – q)•n = 0
• Expand: t d • n q • n = 0
• Solution: t = ( q • n )/( d • n )

p
half-line n d w
If a half-line begins from a point outside a given plane, then it can only hit that plane once (and it might possibly not even hit that plane at all half-line

• Achromatic light: brightness, but no color
• Light can come from point-sources, and light can come from ambient sources
• Incident light shining on the surface of an object can react in three discernable ways:
– By being absorbed
– By being reflected
– By being transmitted (into the interior)

• Besides knowing where a ray of light will hit a surface, we will also need to know whether that ray is reflected or absorbed
• If the ray of light is reflected by a surface, does it travel mainly in one direction? Or does it scatter off in several directions?
• Various surface materials react differently

• Scattering of light (“diffuse reradiation’)
– color may be affected by the surface
• Reflection of light (“specular reflection”)
– mirror-like shininess, color isn’t affected

• Normal vector to the surface at a point
• Direction vector from point to viewer’s eye
• Direction vector from point to light-source