Shape interface

Shapes

Digital Image Synthesis Yung-Yu Chuang

9/27/2005

with slides by Pat Hanrahan

Shapes

• Shape: raw geometry properties of the primitive

• Primitive=Shape+Material

• Source code in core/shape.* and shapes/*

• pbrt provides the following shape plug-ins:

– quadrics: sphere, cone, cylinder, disk, hyperboloid, paraboloid (surface described by quadratic

polynomials in x, y, z) – triangle mesh

– height field – NURBS

– Loop subdivision surface

Shape

class Shape : public ReferenceCounted { public:

<Shape Interface>

const Transform ObjectToWorld, WorldToObject;

const bool reverseOrientation, transformSwapsHandedness;

}

• All shapes are defined in object coordinate space

Shape interface

• BBox ObjectBound(;

• BBox WorldBound() { (can be overridden)

return ObjectToWorld(ObjectBound());

}

• bool CanIntersect() returns whether this shape can do intersection test; if not, the shape must provide

void Refine(vector<Reference<Shape>>&refined)

• bool Intersect(const Ray &ray,

float *tHit, DifferentialGeometry *dg)

• bool IntersectP(const Ray &ray)

Shape interface

• float Area()

• void GetShadingGeometry(

const Transform &obj2world,

const DifferentialGeometry &dg, DifferentialGeometry *dgShading)

• No back culling for that it doesn’t save much for ray tracing and it is not physically correct

for object instancing

Surfaces

• Implicit: F(x,y,z)=0

you can check

• Explicit: (x(u,v),y(u,v),z(u,v)) you can enumerate

• Quadric

[ ]

1

1 =

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

z y x

J I

G D

I H

F C

G F

E B

D C

B A

z y

x

Sphere

• A sphere of radius r at the origin

• Implicit: x²+y²+z²-r²=0

• Parametric: f(θ,ψ) x=rsinθcosψ

y=rsinθsinψ z=rcosθ

mapping f(u,v) over [0,1]² ψ=uψ_max

θ=θ_max+v(θ_max-θ_max)

Sphere

Sphere (construction)

class Sphere: public Shape {

……

private:

float radius;

float phiMax;

float zmin, zmax;

float thetaMin, thetaMax;

}

Sphere(const Transform &o2w, bool ro,

float rad, float zmin, float zmax, float phiMax);

• Bounding box for sphere, only z clipping

Algebraic solution

• Perform in object space, WorldToObject(r, &ray)

• Assume that ray is normalized for a while

2 2

2

2 y z r

x + + =

(

)

(

)

⁽

⁾

2 + Bt + C = 0 At

2 2

2

z y

x d d

d

A = + +

) (

2 d_xo_x d_yo_y d_zo_z

B = + +

2 2

2

2 o o r

o

C = _x + _y + _z −

Step 1

Algebraic solution

A

AC B

t B

2

2 4

0

−

−

= −

A

AC B

t B

2

2 4

1

− +

= −

If (B²-4AC<0) then the ray misses the sphere

Step 2

Step 3

Calculate t₀ and test if t₀<0

Step 4

Calculate t₁ and test if t₁<0

Geometric solution

1. Origin inside? ^o_x^{2 + o2}_y ^{+ o}_z^{2 > r2}

Geometric solution

2. find the closest point, t=-O‧D if t<0 and O outside return false

t

t

Geometric solution

3. find the distance to the origin, d²=O²-t² if s²=r²-d²<0 return false;

t

r

r d

O

s

Geometric solution

4. calculate intersection distance, if (origin outside) then t-s

else t+s

t

r

d

O

s

t

r d

O s

Sphere intersection

• Use algebraic solution, why?

• Consider numeric stability

Partial sphere

• Have to test sphere intersection against clipping parameters

• Partial derivatives

• Area

) 0 , ,

( _max y _max x u

p = −φ φ

∂

∂

) sin ,

sin ,

cos )(

(θ_max θ_min z φ z φ r θ v

p = − −

∂

∂

) ( _{max min}

max r z z

A = φ −

Cylinder φ

φ = u

φ cos r

x =

φ sin r

y =

) (

min

v z z

z

z = + −

Cylinder

Cylinder (intersection)

2 2

2 y r

x + =

(

)

(

)

2 + Bt + C = 0 At

2 2

y

x d

d

A = +

) (

2 d_xo_x d_yo_y

B = +

2 2

2 o r

o

C = _x + _y −

Disk φ

φ = u

φ cos )

) 1

(( v r vr x = −

+

h z =

φ sin )

) 1

(( v r vr

y = −

+

Disk

Disk (intersection)

h-O_z

h t

D_z D ^{z h Oz}

t D

D

= −

)

( _z

z

O D h

t = D −

z z

D O h D

t'= t = −

Other quadrics

Triangle mesh

The most commonly used shape. In pbrt, it can be supplied by users or tessellated from other shapes.

Triangle mesh

class TriangleMesh : public Shape {

…

int ntris, nverts;

int *vertexIndex;

Point *p;

Normal *n;

Vector *s;

float *uvs;

}

x,y,z x,y,z x,y,z x,y,z ^… x,y,z vi[3*i]

vi[3*i+1]

vi[3*i+2]

p

Note that p is stored in world space to save transformations. n and s are in object space.

Triangle mesh

• ObjectBound and WorldBound

Void TriangleMesh::Refine(vector<Reference<Shape>>

&refined) {

for (int i = 0; i < ntris; ++i)

refined.push_back(new Triangle(ObjectToWorld, reverseOrientation, (TriangleMesh *)this, i));

}

1. Intersect ray with plane

2. Check if point is inside triangle

Ray triangle intersection

Algebraic Method 0

:

: ₀

= +

⋅

+

=

d N

P Plane

tV P

P Ray

Substituting for P, we get:

(

)

Solution:

( )

(

)

t P

⋅

+

⋅

= − ⁰

tV P

P =

+

Ray plane intersection

Algebraic Method

( )

( )

FALSE return

d N

P if

N P

d

N Normalize

V V

N

P T

V

P T

V

1 0

1 1 0

1

1 1 2

1

2 2

1 1

<

+

⋅

⋅

−

=

×

=

−

=

−

=

For each side of triangle:

end

Ray triangle intersection I

Parametric Method

( ) ( )

( ) ( )

triangle inside

is P then

and if

T T

T T

P

Compute

0 . 1 0

. 0 0

. 1 0

. 0

: ,

1 3

1 2

≤

≤

≤

≤

− +

−

=

β α

β α

β α

Ray triangle intersection II

Ray triangle intersection III

Fast minimum storage intersection

2 1

)

1

( u v V uV vV tD

O + = − − + + 1

and

0

, v ≥ u + v ≤ u

[

] ^{O V}

v u t V

V V

V

D = −

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

−

−

−

Fast minimum storage intersection

0 1

1

V V

E = − E

= V

− V

T = O − V

E

D

P = × Q = T × E

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

⋅

⋅

⋅

= ⋅

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

D Q

T P

E Q

E v P

u

t

1

1