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About Anamorphic ArtCylinder

The SphereGeneralizations

Java Swing Program

Mathematics Behind Anamorphic Art

Nicole Massarelli (Dr. Don Spickler)

Salisbury University

March 26, 2010

[email protected]

Nicole Massarelli (Dr. Don Spickler) Mathematics Behind Anamorphic Art
http://goforward/http://find/http://goback/


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Java Swing Program

What is Anamorphic Art?

Distorting an image

Only visible fromsingle vantage point

Renaissance andVictorian Era

Earliest examplesfrom Leonardo daVinci

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Java Swing Program

Examples of Anamorphic Art

Julian Beever

http://find/


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Java Swing Program


Gala Contemplating theMediterranean Sea

Salvador Dali


Ab A hi A
http://find/


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Java Swing Program


Istvan Orosz


Ab t A hi A t
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Java Swing Program

General Technique

Goal: Replicate the artistic

process for mirrored surfaces

Find equation for surfaceDefine vector from viewer topixel

Find intersection

Calculate normal vectorFind reflection vector

Determine printingcoordinates


About Anamorphic Art
http://find/


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Java Swing Program

Cylinder

Set-up

Choose viewer

position (x0, y0, z0)

Write equation forcylinder x2 + y2 = r2

Place image inside

cylinderChoose arbitrary pixelon image (xp, yp, zp)


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Java Swing Program

Cylinder

Finding the Intersection

Write vparametrically

xv = xp +(x0 xp) t = xp + at

yv = yp +(y0 yp) t = yp + bt

zv = zp +(z0 zp) t = zp + ct

Plug parametricequations into

cylinder equationNicole Massarelli (Dr. Don Spickler) Mathematics Behind Anamorphic Art

http://find/


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Java Swing Program

Cylinder


(a2 + b2)t2 + 2(xpa + ypb)t + x2p

+ y2p r2 = 0

Solve using quadratic formula

Plug t into parametric equations of v


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Java Swing Program

Cylinder


Determine pointclosest to viewer

Distance formula

Point of intersection(x1, y1, z1)


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pCylinder


Java Swing Program

Cylinder

Calculating the Normal

Vector

Find point on z-axisat same height as(x1, y1, z1)

(0, 0, z1)

Calculate vectorbetween these points




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pCylinder


Java Swing Program

Cylinder

Find Reflection Vector

Project v onto nn = projnv =

nv|n|2

n

Solve for a usingvector addition

a = n

vr = v + 2a


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CylinderThe Sphere

GeneralizationsJava Swing Program

Cylinder

Printing the Pixel

Write r parametrically

xr = x1 + dtyr = y1 + et

zr = z1 + ft

Determine where r

intersects xy-planeSolve zr = 0 for t

Printing coordinates(xr, yr, 0)


About Anamorphic ArtC
http://find/


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CylinderThe Sphere


Sphere

Viewing Image on

Mirrored Sphere

Set sphere on

pedestal inside boxFix viewer position

Follow generalalgorithm

Key DifferencesFinding normalvectorChecking eachface for

intersection pointsNicole Massarelli (Dr. Don Spickler) Mathematics Behind Anamorphic Art

About Anamorphic ArtC li d
http://find/


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CylinderThe Sphere


Other Surfaces

Generalization

Triangulation

methods for meshsurfaces- Platonicsolids

Key Differences

Barycentric

coordinates usedto find intersectionNormal vector iscross product oftwo edges


About Anamorphic ArtC li d
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CylinderThe Sphere


Curvature

Convex Surfaces

We can only useconvex surfaces

Concave surfaces givedouble images

Proof intwo-dimensional case


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CylinderThe Sphere


Anamorphic Art Converter




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CylinderThe Sphere


Sample Program Output


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CylinderThe Sphere


Cylinder Model


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CylinderThe Sphere


Link to Program

http://facultyfp.salisbury.edu/despickler/personal/

CylinderReflectionGenerator.asp

Created under the direction of Dr. Don Spickler




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yThe Sphere


References

Differential Geometry by John Oprea

Linear Algebra by David C. LayJava Swing by Marc Loy, Robert Eckstein et al

Calculus: Early Transcendentals by James Stewart

3D Computer Graphics:A Mathematical Introduction with

OpenGL by Samuel R. Buss

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