GR2 Advanced Computer Graphics AGR Lecture 13 An Introduction to Ray Tracing

Ray Tracing Ray tracing is an alternative rendering approach to the polygon projection, shading and z-buffer way of working It is capable of high quality images through taking account of global illumination

Ray Tracing Example

Firing Rays The view plane is marked with a grid corresponding pixel positions on view plane camera light The view plane is marked with a grid corresponding to pixel positions on screen. A ray is traced from the camera through each pixel in turn.

Finding Intersections pixel positions on view plane camera light We calculate the intersection of the ray with all objects in the scene. The nearest inter- section will be the visible surface for that ray

Intensity Calculation pixel positions on view plane camera light The intensity at this nearest intersection is found from the usual Phong reflection model. Stopping at this point gives us a very simple rendering method. Often called: ray casting

Phong Reflection Model light source N R eye L  V  surface I() = Ka()Ia() + ( Kd()( L . N ) + Ks( R . V )n ) I*() / dist dist = distance attenuation factor Here V is direction of incoming ray, N is normal, L is direction to light source, and R is direction of perfect spec. reflection. Note: R.V calculation replaced by H.N for speed - H = (L+V)/2

Intensity Calculation pixel positions on view plane camera light Thus Phong reflection model gives us intensity at point based on a local model: I = I local where I local = I ambient + I diffuse + I specular Intensity calculated separately for red, green, blue

Shadows To determine if the point is in shadow, we can fire a ray at pixel positions on view plane camera light To determine if the point is in shadow, we can fire a ray at the light source(s) - in direction L. If the ray intersects another object, then point is in shadow from that light. In this case, we just use ambient component: I local = I ambient Otherwise the point is fully lit. shadow ray

Reflected Ray R1 R1 = V - 2 (V.N) N Ray tracing models pixel positions on view plane camera light Ray tracing models reflections by looking for light coming in along a secondary ray, making a perfect specular reflection. R1 = V - 2 (V.N) N R1

Reflected Ray- Intersection and Recursion pixel positions on view plane camera light We calculate the intersection of this reflected ray, with all objects in the scene. The intensity at the nearest intersection point is calculated, and added as a contribution attenuated by distance. This is done recursively.

Reflected Ray - Intensity Calculation pixel positions on view plane camera light The intensity calculation is now: I = I local + k r * I reflected Here I reflected is calculated recursively k r is a reflection coefficient (similar to k s )

Ray Termination Rays terminate: - on hitting diffuse surface pixel positions on view plane camera Rays terminate: - on hitting diffuse surface - on going to infinity - after several reflections - why?

Transmitted Ray If the object is semi- transparent, we also pixel positions on view plane camera If the object is semi- transparent, we also need to take into account refraction light T1 Thus we follow also transmitted rays, eg T1.

Refraction r i r i Snell’s Law: sin  r = ( i /  r ) * sin  i Change in direction is determined by the refractive indices of the materials, i and r T r N i V Snell’s Law: sin  r = ( i /  r ) * sin  i T = ( i /  r ) V - ( cos  r - ( i /  r ) cos  i ) N

Refraction Contribution The contribution due to transmitted light is taken as: kt * It(  ) where kt is the transmission coefficient It(  ) is the intensity of transmitted light, again calculated separately for red, green, blue

Intensity Calculation pixel positions on view plane camera The intensity calculation is now: I = I local + k r * I reflected + k t * I transmitted I reflected and I transmitted are calculated recursively

Binary Ray Tracing Tree S4 pixel positions on view plane camera light S1 T1 R1 S2 S3 R1 S2 S3 T1 S1 S4 The intensity is calculated by traversing the tree and accumulating intensities

Calculating Ray Intersections A major part of ray tracing calculation is the intersection of ray with objects Two important cases are: sphere polygon

Ray - Sphere Intersection Let camera position be (x1, y1, z1) Let pixel position be (x2, y2, z2) Any point on ray is: x(t) = x1 + t * (x2 - x1) = x1 + t * i y(t) = y1 + t * (y2 - y1) = y1 + t * j z(t) = z1 + t * (z2 - z1) = z1 + t * k As t increases from zero, x(t), y(t), z(t) traces out the line from (x1, y1, z1) through (x2, y2, z2). Equation of sphere, centre (l, m, n) and radius r: (x - l)2 + (y - m)2 + (z - n)2 = r2

Ray - Sphere Intersection Putting the parametric equations for x(t), y(t), z(t) in the sphere equation gives a quadratic in t: at2 + bt + c = 0 Exercise: write down a, b, c in terms of i, j, k, l, m, n, x1, y1, z1 Solving for t gives the intersection points: b2 - 4ac < 0 no intersections b2 - 4ac = 0 ray is tangent b2 - 4ac > 0 two intersections, we want smallest positive

Ray - Polygon Intersection Equation of ray: x(t) = x1 + t * i y(t) = y1 + t * j z(t) = z1 + t * k Equation of plane: ax + by + cz + d = 0 Intersection: t = - (ax1 + by1 + cz1 + d) / (ai + bj + ck) Need to check intersection within the extent of the polygon.

Ray Tracing - Limitations Ray tracing in its basic form is computationally intensive Much research has gone into increasing the efficiency and this will be discussed in next lecture.

Ray Tracing Example

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Acknowledgements As ever, thanks to Alan Watt for the excellent images