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Intro CG @ NCSU

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curves

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Intro CG @ NCSU

Ben Watson

cgclass.csc.ncsu.edu

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curves

go-vxlab.csc.ncsu.edu/

curvesslides

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why?

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history

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math!

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P₁

Q(t) = P₁ + t(P₂- P₁)

P₂

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Q(t) = P₁ + t(P₂- P₁)

t = 1

t = 0

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Q(0)

Q(t) = P₁ + t(P₂- P₁)

Q(1)

Q(t)

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Q(t) = t³ (a G₁ + b G₂ + c G₃ + d G₄)

+ t² (e G₁ + f G₂ + g G₃ + h G₄)

+ t (i G₁ + j G₂ + k G₃ + l G₄)

+ (m G₁ + n G₂ + o G₃ + p G₄)

G₄

G₃

G₁

G₂

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Q(t) = t³ (a G₁ + b G₂ + c G₃ + d G₄)

+ t² (e G₁ + f G₂ + g G₃ + h G₄)

+ t (i G₁ + j G₂ + k G₃ + l G₄)

+ (m G₁ + n G₂ + o G₃ + p G₄)

t = 1

t = 0

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Q(t) = t³ (a G₁ + b G₂ + c G₃ + d G₄)

+ t² (e G₁ + f G₂ + g G₃ + h G₄)

+ t (i G₁ + j G₂ + k G₃ + l G₄)

+ (m G₁ + n G₂ + o G₃ + p G₄)

Q(1)

Q(0)

Q(t)

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Q(t) = t³ (a G₁ + b G₂ + c G₃ + d G₄)

+ t² (e G₁ + f G₂ + g G₃ + h G₄)

+ t (i G₁ + j G₂ + k G₃ + l G₄)

+ 1 (m G₁ + n G₂ + o G₃ + p G₄)

Q(t) = T M G

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Q(t) = T M G

geometry vector

G₄

G₃

G₁

G₂

G₁

G₂

G₃

G₄

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Q(t) = T M G

basis matrix

a b c d

e f g h

i j k l

m n o p

G₁

G₂

G₃

G₄

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Q(t) = T M G

(bezier) basis matrix

-1 3 -3 1

3 -6 3 0

-3 3 0 0

1 0 0 0

G₁

G₂

G₃

G₄

t

geom weight

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Q(t) = T M G

parameter vector

a b c d

e f g h

i j k l

m n o p

G₁

G₂

G₃

G₄

t³ t² t 1

t = 1

t = 0

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control

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locality

bad locality

high order curve

good locality

b-spline

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locality

pomax.github.io/bezierinfo/ ...

... #reordering

#bsplines

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join continuity

C1

same

loc & slope

C0

same

location

C2

same

loc, slope, 2^nd deriv

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join continuity

math.hws.edu/graphicsbook/demos/c2/cubic-bezier.html

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convex hull

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convex hull

bl.ocks.org/mbostock/raw/4342190/

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interpolate vs approximate

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interpolate vs approximate

mbostock.github.io/protovis/ex/splines.html

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transform invariance

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↖

ask your questions

BREAK TIME

listen to

tunes

↓

♬

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types

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Hermite

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Hermite

www.desmos.com/calculator/5knm5tkr8m

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Hermite

	locality	continuity	convex hull	invariance	interpolation
Hermite

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Hermite

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗

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Hermite

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗

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Hermite

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗

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Hermite

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine

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Hermite

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints

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Bezier

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Bezier

javascript.info/bezier-curve

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Bezier

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗

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Bezier

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗

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Bezier

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓

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Bezier

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine

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Bezier

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints

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cardinal

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cardinal

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cardinal

www.ibiblio.org/e-notes/Splines/cardinal.html

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cardinal

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓

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cardinal

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1

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cardinal

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗

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cardinal

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine

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cardinal

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓

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B-spline

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B-spline

pomax.github.io/bezierinfo/#bsplines

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B-spline

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓

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B-spline

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2

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B-spline

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓

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B-spline

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine

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B-spline

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗

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NURBS

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NURBS

nurbscalculator.in

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NURBS

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗
NURBS	✓

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NURBS

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗
NURBS	✓	C2

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NURBS

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗
NURBS	✓	C2	✓

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NURBS

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗
NURBS	✓	C2	✓	perspective

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NURBS

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗
NURBS	✓	C2	✓	perspective	as needed

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NURBS

	locality	continuity	convex hull	invariance	interpolation
Hermite	✗	✗	✗	affine	endpoints
Bezier	✗	✗	✓	affine	endpoints
cardinal	✓	C1	✗	affine	✓
B-spline	✓	C2	✓	affine	✗
NURBS	✓	C2	✓	perspective	as needed

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patches

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www.ibiblio.org/e-notes/Splines/bezier3d.html

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subdivision

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www.ibiblio.org/e-notes/Splines/adaptive.html

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go deeper with readings...

sites.google.com/view/cgwiki-ncsu/topics/curves

post reactions and... we might discuss them next time

log them and... get reading credit

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...and with videos

cgclass.csc.ncsu.edu/p/topic-notes.html#curves

(youtube)

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summaries, questions & thoughts @

go-vxlab.csc.ncsu.edu / curvesdiscussion

post them and... we’ll discuss them shortly

log them and... get participation credit

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