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Dr. Scott Schaefer

Intersecting Simple Surfaces

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Types of Surfaces

Infinite Planes Polygons

Convex Ray Shooting Winding Number

Spheres Cylinders

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Infinite Planes

Defined by a unit normal n and a point o

0)( oxn

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Infinite Planes

Defined by a unit normal n and a point o

0)( oxntvptL )(

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Infinite Planes

Defined by a unit normal n and a point o

0)( oxntvptL )(

0)( otvpn

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Infinite Planes

Defined by a unit normal n and a point o

0)( oxntvptL )(

)( pontvn

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Infinite Planes

Defined by a unit normal n and a point o

0)( oxntvptL )(

vn

pont

)(

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Infinite Planes

Defined by a unit normal n and a point o

0)( oxntvptL )(

vn

ponvp

)(

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Polygons

Intersect infinite plane containing polygon Determine if point is inside polygon

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Polygons

Intersect infinite plane containing polygon Determine if point is inside polygon

How do we know if a point is inside a polygon?

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Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Check if point on same side of all edges

Point Inside Convex Polygon

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Point Inside Convex Polygon

X iP

1iP sign same bemust

)(

)(

1T

i

Ti

T

XP

XP

N

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Point Inside Polygon Test

Given a point, determine

if it lies inside a polygon

or not

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Ray Test

Fire ray from point Count intersections

Odd = inside polygonEven = outside polygon

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Problems With Rays

Fire ray from point Count intersections

Odd = inside polygonEven = outside polygon

ProblemsRay through vertex

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Problems With Rays

Fire ray from point Count intersections

Odd = inside polygonEven = outside polygon

ProblemsRay through vertex

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Problems With Rays

Fire ray from point Count intersections

Odd = inside polygonEven = outside polygon

ProblemsRay through vertexRay parallel to edge

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

One winding = inside

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

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A Better Way

zero winding = outside

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Requirements

Oriented edges Edges can be processed

in any order

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Computing Winding Number

Given unit normal n =0 For each edge (p1, p2)

If , then inside

1p

2px

xpxp

xpxp

xpxp

xpxpn

21

211

21

21 )()(cos

)()(

))()((

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Advantages

Extends to 3D! Numerically stable Even works on models

with holes (sort of) No ray casting

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Intersecting Spheres

Three possible casesZero intersections: miss the sphereOne intersection: hit tangent to sphereTwo intersections: hit sphere on front and

back side

How do we distinguish these cases?

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Intersecting Spheres

0)()()( 2 rcxcxxF

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Intersecting Spheres

0)()()( 2 rcxcxxF

0)()())(( 2 rctvpctvptLF

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Intersecting Spheres

0)()()( 2 rcxcxxF

0)()())(( 2 rctvpctvptLF

0)()()(2)())(( 22 rcpcptcpvtvvtLF

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Intersecting Spheres

is quadratic in t0))(( tLF

0)()()(2)())(( 22 rcpcptcpvtvvtLF

a b c

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Intersecting Spheres

is quadratic in t

Solve for t using quadratic equation

If , no intersection If , one intersection Otherwise, two intersections

0))(( tLF

0)()()(2)())(( 22 rcpcptcpvtvvtLF

a b c

a

acbbt

2

42

042 acb

042 acb

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Normals of Spheres

0)()()( 2 rcxcxxF

cxxF )(

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r)(tL

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r

1.Perform an orthogonal projection to the plane defined by C, A on the line L(t) and intersect with circle in 2D

)(ˆ tL

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r)(tL

2.Substitute t parameters from 2D intersection to 3D line equation

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r)(tL

3.Normal of 2D circle is the same normal of cylinder at point of intersection

N

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r

PN

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Infinite Cylinders

Defined by a center point C, a unit axis direction A and a radius r

A

C

r

PN

(( ) )P C P C A AN

r