The photorealism quest An introduction to raytracing The

The photorealism quest An introduction to ray-tracing

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

Example: a spherical object • 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

Describing the sphere • 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 = ( cx, cy, cz ) be the sphere’s center • If w = ( wx, wy, wz ) 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 }

Describing a ray • • • A ray is a geometrical half-line in space It has a beginning point: b = ( bx, by, bz ) It has a direction-vector: d = ( dx, dy, dz ) It can be described with a parameter t ≥ 0 If w = ( wx, wy, wz ) is any point in space, then w will be on the half-line just in case w = b + td for some choice of the parameter t ≥ 0.

Computing the ‘hit’ time • 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

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

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

Interpretations half-line w 2 w 1 b c A half-line that begins inside the sphere will only ‘hit’ the spgere’s surface once half-line

Second example: A planar surface • 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 }

When does ray ‘hit’ plane? • • Plane: (w – p) • n = 0 Half-line: w = b + td, t≥ 0 Substitution: (b + td – p) • n = 0 Replacement: Let q = p – b Simplification: (td – q) • n = 0 Expand: td • n - q • n = 0 Solution: t = (q • n)/(d • n)

Interpretations half-line n p 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 • 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)

Illumination • 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

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

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