07 - Designing Interpolating Curves

CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

07 - Designing Interpolating Curves

Acknowledgement: Olga Sorkine-Hornung, Alexander Sorkine-Hornung, Ilya Baran


How do we model shapes?

Delcam Plc. [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons


Building blocks: curves and surfaces

By Wojciech mula at Polish Wikipedia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=9853555


A modeling session

• Demo with Keynote for 2D

• Demo with Blender for 3D


Modeling curves

• We need mathematical concepts to characterize the desired curve properties

• Notions from curve geometry help with designing user interfaces for curve creation and editing


2D parametric curve• must be continuous


A curve can be parameterized in many different ways


Tangent vector

Parametrizat

ion-indepen

dent!


Arc length• How long is the curve between and ? How far does the particle

travel?

• Speed is , so:

• Speed is nonnegative, so is non-decreasing


Arc length parameterization• Every curve has a natural parameterization:


Arc length parameterization• Every curve has a natural parameterization:

• Isometry between parameter domain and curve

• Tangent vector is unit-length:


Curvature• How much does the curve turn per unit ?


Curvature profile

• Given , we can get up to a constant by integration. Integrating

reconstructs the curve up to rigid motion.


Curvature of a circle• Curvature of a circle:

Osculating circle

= Unit Normal


Frenet Frame


Curvature normal• Points inward

• useful for evaluating curve quality


SmoothnessTwo kinds, parametric and geometric:

Parametrization-Independent


Smoothness example


Recap on parametric curves


Turning• Angle from start tangent to end tangent:

• If curve is closed, the tangent at the beginning is the same as the tangent at the end


Turning numbers

• measures how many full turns the tangent makes.

1 –1 2 0


Space curves (3D)• In 3D, many vectors are orthogonal to T

• are the “Frenet frame”

• is torsion: non-planarity

N

T

B


Designing Curves:Polynomials and Interpolation


Basic idea for curve design• User gives us points. We need to connect the dots in a smooth way.

• The dots stay as “handles” on the curve. User can move the dots and the curve follows.


Keynote Demo


Blender Demo


Polynomial curves• Polynomials

• For degree we need points (“coefficients”)


Polynomials• Parametric form with polynomials

• Control shape?


Polynomials• Parametric form with polynomials

• Control shape?

• Monomial coefficients are not intuitive


Interpolation (1D)• Interpolate control points • Find polynomial

with



with • Solve

Vandermonde matrix



with • Solve


Interpolation (1D)• Polynomial fitting can be done explicitly:

• Check that

• Products are called Lagrange polynomials


Example of 1D Interpolation


Recap

• Monomial coefficients unintuitive

• Lagrange interpolation is better, but there is a global effect

• The possible curves are the same: just cubics


Polynomials form a vector space• You can add and subtract them

• You can multiply them by real numbers

• These operations follow the usual rules • Associativity, commutativity, inverses for addition • Distributivity for scalar multiplication • Etc.

• Anything you can do with vectors in general, you can do with polynomials.


Monomial basis for cubics:


Interpolation basis for :


Remark

• Basis change is just a matrix multiplication

• We need to find a good basis for intuitively designing curves


Interpolation in 2D• We now have pairs we want to interpolate:

• How to choose ’s at which to specify ?

• Parameter space matters!


Runge’s phenomenon


Splines• Paste together low-degree polynomials

CubicCub

ic

Cubic

Cubic Cub

ic


How do we get smoothness?• With Lagrange polynomials, it’s hard to get tangents to match up


Hermite Basis• Instead of four points, specify two points and two derivatives:

Cubic polynomials of the form

p(t0) = p0

p(t1) = p1

p0(t0) = m0

p0(t1) = m1


Hermite Basis• Instead of four points, specify two points and two derivatives:


Catmull-Rom splines• If no tangents are explicitly specified – get them from the input

points!


Next time

• Approximating Curves • Bezier splines • B-splines


References

Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley

Chapter 15

https://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=Steve+Marschner&search-alias=books&field-author=Steve+Marschner&sort=relevancerank

https://www.amazon.com/Peter-Shirley/e/B00IZTGJ9O/ref=dp_byline_cont_book_2

07 - Designing Interpolating Curves

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