website articles
intersectors

### Intro

This is a collection of ray-primitive intersectors for some common primitive types, which you can use for raytracing. My game is always to derive these myself, so there might be errors or precission errors (in particular the Torus), althogh all come with actual running examples. Note that all the intersectors listed allow to specify the primitive in arbitrary locations and orientations, except for the Box, Ellipsoid and Torus, which are specifically located at the origin and aligned to the XYZ axis - for those two primitives you'll need to transform the ray to the primitive's spaces. I'll add more over time. Also, a nice feature of the capped cone and capsule intersectors is that they only check against one of the caps, even though the primitive has two.

In all the examples bellow, ro is the ray origin, and rd is the ray direction. The functions always return the distance to the closest intersection, and sometimes some additional information (second intersection, normal or barycentric coordinates). Most primitives are centeted at the origin (you'll need to transform the ray origin and direction accordingly), but some of the functions above accept an arbitrary primitive location and orientation when such implementation is faster than performing a ray transformation.

Sphere
live example
// sphere of size ra centered at point ce vec2 sphIntersect( in vec3 ro, in vec3 rd, in vec3 ce, float ra ) { vec3 oc = ro - ce; float b = dot( oc, rd ); float c = dot( oc, oc ) - ra*ra; float h = b*b - c; if( h<0.0 ) return vec2(-1.0); // no intersection h = sqrt( h ); return vec2( -b-h, -b+h ); }

Plane
// plane degined by p (p.xyz must be normalized) float plaIntersect( in vec3 ro, in vec3 rd, in vec4 p ) { return -(dot(ro,p.xyz)+p.w)/dot(rd,p.xyz); }

// cylinder defined by extremes pa and pb, and radious ra vec4 cylIntersect( in vec3 ro, in vec3 rd, in vec3 pa, in vec3 pb, float ra ) { vec3 ca = pb-pa; vec3 oc = ro-pa; float caca = dot(ca,ca); float card = dot(ca,rd); float caoc = dot(ca,oc); float a = caca - card*card; float b = caca*dot( oc, rd) - caoc*card; float c = caca*dot( oc, oc) - caoc*caoc - ra*ra*caca; float h = b*b - a*c; if( h<0.0 ) return vec4(-1.0); //no intersection h = sqrt(h); float t = (-b-h)/a; // body float y = caoc + t*card; if( y>0.0 && y<caca ) return vec4( t, (oc+t*rd-ca*y/caca)/ra ); // caps t = (((y<0.0)?0.0:caca) - caoc)/card; if( abs(b+a*t)<h ) return vec4( t, ca*sign(y)/caca ); return vec4(-1.0); //no intersection }

### Cylinder

// inifintie cylinder defined by a base point cb, a normalized axis ca and a radious cr vec2 cylIntersect( in vec3 ro, in vec3 rd, in vec3 cb, in vec3 ca, float cr ) { vec3 oc = ro - cb; float card = dot(ca,rd); float caoc = dot(ca,oc); float a = 1.0 - card*card; float b = dot( oc, rd) - caoc*card; float c = dot( oc, oc) - caoc*caoc - cr*cr; float h = b*b - a*c; if( h<0.0 ) return vec2(-1.0); //no intersection h = sqrt(h); return vec2(-b-h,-b+h)/a; }

Capped cone
live example
// cone defined by extremes pa and pb, and radious ra and rb Only one square root and one division is emplyed in the worst case. dot2(v) is dot(v,v) vec4 coneIntersect( in vec3 ro, in vec3 rd, in vec3 pa, in vec3 pb, in float ra, in float rb ) { vec3 ba = pb - pa; vec3 oa = ro - pa; vec3 ob = ro - pb; float m0 = dot(ba,ba); float m1 = dot(oa,ba); float m2 = dot(rd,ba); float m3 = dot(rd,oa); float m5 = dot(oa,oa); float m9 = dot(ob,ba); //caps if( m1<0.0 ) { if( dot2(oa*m2-rd*m1)<(ra*ra*m2*m2) ) // delayed division return vec4(-m1/m2,-ba*inversesqrt(m0)); } else if( m9>0.0 ) { float t = -m9/m2; // NOT delayed division if( dot2(ob+rd*t)<(rb*rb) ) return vec4(t,ba*inversesqrt(m0)); } // body float rr = ra - rb; float hy = m0 + rr*rr; float k2 = m0*m0 - m2*m2*hy; float k1 = m0*m0*m3 - m1*m2*hy + m0*ra*(rr*m2*1.0 ); float k0 = m0*m0*m5 - m1*m1*hy + m0*ra*(rr*m1*2.0 - m0*ra); float h = k1*k1 - k2*k0; if( h<0.0 ) return vec4(-1.0); //no intersection float t = (-k1-sqrt(h))/k2; float y = m1 + t*m2; if( y<0.0 || y>m0 ) return vec4(-1.0); //no intersection return vec4(t, normalize(m0*(m0*(oa+t*rd)+rr*ba*ra)-ba*hy*y)); }

Ellipsoid
live example
// ellipsoid centered at the origin with radii ra float eliIntersect( in vec3 ro, in vec3 rd, in vec3 ra ) { vec3 ocn = ro/ra; vec3 rdn = rd/ra; float a = dot( rdn, rdn ); float b = dot( ocn, rdn ); float c = dot( ocn, ocn ); float h = b*b - a*(c-1.0); if( h<0.0 ) return -1.0; //no intersection h = sqrt(h); return vec2(-b-h,-b+h)/a; } vec3 eliNormal( in vec3 pos, in vec3 ra ) { return normalize( pos/ra ); }

Triangle
live example
// triangle degined by vertices v0, v1 and v2 vec3 triIntersect( in vec3 ro, in vec3 rd, in vec3 v0, in vec3 v1, in vec3 v2 ) { vec3 v1v0 = v1 - v0; vec3 v2v0 = v2 - v0; vec3 rov0 = ro - v0; vec3 n = cross( v1v0, v2v0 ); vec3 q = cross( rov0, rd ); float d = 1.0/dot( rd, n ); float u = d*dot( -q, v2v0 ); float v = d*dot( q, v1v0 ); float t = d*dot( -n, rov0 ); if( u<0.0 || u>1.0 || v<0.0 || (u+v)>1.0 ) t = -1.0; return vec3( t, u, v ); }

Ellipse
live example
// ellipse defined by its center c and its two radii u and v vec3 ellIntersect( in vec3 ro, in vec3 rd, vec3 c, vec3 u, vec3 v ) { vec3 q = ro - c; vec3 n = cross(u,v); float t = -dot(n,q)/dot(rd,n); float r = dot(u,q + rd*t); float s = dot(v,q + rd*t); if( r*r+s*s>1.0 ) return vec3(-1.0); //no intersection return vec3(t,s,r); } vec3 ellNormal( in vec2 u, vec2 v ) { return normalize( cross(u,v) ); }

Torus
live example
float torIntersect( in vec3 ro, in vec3 rd, in vec2 tor ) { float po = 1.0; float Ra2 = tor.x*tor.x; float ra2 = tor.y*tor.y; float m = dot(ro,ro); float n = dot(ro,rd); float k = (m + Ra2 - ra2)/2.0; float k3 = n; float k2 = n*n - Ra2*dot(rd.xy,rd.xy) + k; float k1 = n*k - Ra2*dot(rd.xy,ro.xy); float k0 = k*k - Ra2*dot(ro.xy,ro.xy); if( abs(k3*(k3*k3-k2)+k1) < 0.01 ) { po = -1.0; float tmp=k1; k1=k3; k3=tmp; k0 = 1.0/k0; k1 = k1*k0; k2 = k2*k0; k3 = k3*k0; } float c2 = k2*2.0 - 3.0*k3*k3; float c1 = k3*(k3*k3-k2)+k1; float c0 = k3*(k3*(c2+2.0*k2)-8.0*k1)+4.0*k0; c2 /= 3.0; c1 *= 2.0; c0 /= 3.0; float Q = c2*c2 + c0; float R = c2*c2*c2 - 3.0*c2*c0 + c1*c1; float h = R*R - Q*Q*Q; if( h>=0.0 ) { h = sqrt(h); float v = sign(R+h)*pow(abs(R+h),1.0/3.0); // cube root float u = sign(R-h)*pow(abs(R-h),1.0/3.0); // cube root vec2 s = vec2( (v+u)+4.0*c2, (v-u)*sqrt(3.0)); float y = sqrt(0.5*(length(s)+s.x)); float x = 0.5*s.y/y; float r = 2.0*c1/(x*x+y*y); float t1 = x - r - k3; t1 = (po<0.0)?2.0/t1:t1; float t2 = -x - r - k3; t2 = (po<0.0)?2.0/t2:t2; float t = 1e20; if( t1>0.0 ) t=t1; if( t2>0.0 ) t=min(t,t2); return t; } float sQ = sqrt(Q); float w = sQ*cos( acos(-R/(sQ*Q)) / 3.0 ); float d2 = -(w+c2); if( d2<0.0 ) return -1.0; float d1 = sqrt(d2); float h1 = sqrt(w - 2.0*c2 + c1/d1); float h2 = sqrt(w - 2.0*c2 - c1/d1); float t1 = -d1 - h1 - k3; t1 = (po<0.0)?2.0/t1:t1; float t2 = -d1 + h1 - k3; t2 = (po<0.0)?2.0/t2:t2; float t3 = d1 - h2 - k3; t3 = (po<0.0)?2.0/t3:t3; float t4 = d1 + h2 - k3; t4 = (po<0.0)?2.0/t4:t4; float t = 1e20; if( t1>0.0 ) t=t1; if( t2>0.0 ) t=min(t,t2); if( t3>0.0 ) t=min(t,t3); if( t4>0.0 ) t=min(t,t4); return t; } vec3 torNormal( in vec3 pos, vec2 tor ) { return normalize( pos*(dot(pos,pos)-tor.y*tor.y - tor.x*tor.x*vec3(1.0,1.0,-1.0))); }

Sphere 4
live example
float sph4Intersect( in vec3 ro, in vec3 rd, in float ra ) { float r2 = ra*ra; vec3 d2 = rd*rd; vec3 d3 = d2*rd; vec3 o2 = ro*ro; vec3 o3 = o2*ro; float ka = 1.0/dot(d2,d2); float k3 = ka* dot(ro,d3); float k2 = ka* dot(o2,d2); float k1 = ka* dot(o3,rd); float k0 = ka*(dot(o2,o2) - r2*r2); float c2 = k2 - k3*k3; float c1 = k1 + 2.0*k3*k3*k3 - 3.0*k3*k2; float c0 = k0 - 3.0*k3*k3*k3*k3 + 6.0*k3*k3*k2 - 4.0*k3*k1; float p = c2*c2 + c0/3.0; float q = c2*c2*c2 - c2*c0 + c1*c1; float h = q*q - p*p*p; if( h<0.0 ) return -1.0; //no intersection float sh = sqrt(h); float s = sign(q+sh)*pow(abs(q+sh),1.0/3.0); // cuberoot float t = sign(q-sh)*pow(abs(q-sh),1.0/3.0); // cuberoot vec2 w = vec2( s+t,s-t ); vec2 v = vec2( w.x+c2*4.0, w.y*sqrt(3.0) )*0.5; float r = length(v); return -abs(v.y)/sqrt(r+v.x) - c1/r - k3; } vec3 sph4Normal( in vec3 pos ) { return normalize( pos*pos*pos ); }

Goursat
live example
float gouIntersect( in vec3 ro, in vec3 rd, in float ka, float kb ) { float po = 1.0; vec3 rd2 = rd*rd; vec3 rd3 = rd2*rd; vec3 ro2 = ro*ro; vec3 ro3 = ro2*ro; float k4 = dot(rd2,rd2); float k3 = dot(ro ,rd3); float k2 = dot(ro2,rd2) - kb/6.0; float k1 = dot(ro3,rd ) - kb*dot(rd,ro)/2.0; float k0 = dot(ro2,ro2) + ka - kb*dot(ro,ro); k3 /= k4; k2 /= k4; k1 /= k4; k0 /= k4; float c2 = k2 - k3*(k3); float c1 = k1 + k3*(2.0*k3*k3-3.0*k2); float c0 = k0 + k3*(k3*(c2+k2)*3.0-4.0*k1); if( abs(c1) < 0.1*abs(c2) ) { po = -1.0; float tmp=k1; k1=k3; k3=tmp; k0 = 1.0/k0; k1 = k1*k0; k2 = k2*k0; k3 = k3*k0; c2 = k2 - k3*(k3); c1 = k1 + k3*(2.0*k3*k3-3.0*k2); c0 = k0 + k3*(k3*(c2+k2)*3.0-4.0*k1); } c0 /= 3.0; float Q = c2*c2 + c0; float R = c2*c2*c2 - 3.0*c0*c2 + c1*c1; float h = R*R - Q*Q*Q; if( h>0.0 ) // 2 intersections { h = sqrt(h); float s = sign(R+h)*pow(abs(R+h),1.0/3.0); // cube root float u = sign(R-h)*pow(abs(R-h),1.0/3.0); // cube root float x = s+u+4.0*c2; float y = s-u; float ks = x*x + y*y*3.0; float k = sqrt(ks); float t = -0.5*po*abs(y)*sqrt(6.0/(k+x)) - 2.0*c1*(k+x)/(ks+x*k) - k3; return (po<0.0)?1.0/t:t; } // 4 intersections float sQ = sqrt(Q); float w = sQ*cos(acos(-R/(sQ*Q))/3.0); float d2 = -w - c2; if( d2<0.0 ) return -1.0; //no intersection float d1 = sqrt(d2); float h1 = sqrt(w - 2.0*c2 + c1/d1); float h2 = sqrt(w - 2.0*c2 - c1/d1); float t1 = -d1 - h1 - k3; t1 = (po<0.0)?1.0/t1:t1; float t2 = -d1 + h1 - k3; t2 = (po<0.0)?1.0/t2:t2; float t3 = d1 - h2 - k3; t3 = (po<0.0)?1.0/t3:t3; float t4 = d1 + h2 - k3; t4 = (po<0.0)?1.0/t4:t4; float t = 1e20; if( t1>0.0 ) t=t1; if( t2>0.0 ) t=min(t,t2); if( t3>0.0 ) t=min(t,t3); if( t4>0.0 ) t=min(t,t4); return t; } vec3 gouNormal( in vec3 pos, float ka, float kb ) { return normalize( 4.0*pos*pos*pos - 2.0*pos*kb ); }