Intro to Splines

CS 4620 Lecture 14

Cornell CS4620 Fall 2020 Steve Marschner • 1 Motivation: smoothness

• In many applications we need smooth shapes – that is, without discontinuities [Boeing] • So far we can make – things with corners (lines, triangles, squares, rectangles, …) – circles, ellipses, other special shapes (only get you so far!)

Cornell CS4620 Fall 2020 Steve Marschner • 2 Classical approach

• Pencil-and-paper draftsmen also needed smooth curves • Origin of “spline:” strip of ﬂexible metal – held in place by pegs or weights to constrain shape – traced to produce smooth contour

Cornell CS4620 Fall 2020 Steve Marschner • 3 Translating into usable math

• Smoothness – in drafting spline, comes from physical curvature minimization – in CG spline, comes from choosing smooth functions • usually low-order polynomials • Control – in drafting spline, comes from ﬁxed pegs – in CG spline, comes from user-speciﬁed control points

Cornell CS4620 Fall 2020 Steve Marschner • 4 Piecewise polynomial functions

p0(t) t0 ≤ t < t1 p (t) t ≤ t < t f(t) = 1 1 2 ⋮ pn−1(t) tn−1 ≤ t ≤ tn

spline segments join at knots

knot values Cornell CS4620 Fall 2020 Steve Marschner • 5 Spline segment

• deﬁnes the value of a spline over an interval between adjacent knot values • is a polynomial with coefﬁcients that depend linearly on one or more controls • type and meaning of controls differs among types of spline

Cornell CS4620 Fall 2020 Steve Marschner • 6 Spline continuity pk−1(tk) ≠ pk(tk)

• Knots are transitions between segments – match values for continuity (C0) – match derivatives for smoothness (C1) pk−1(tk) = pk(tk) 0 C pk′−1(tk) ≠ pk′(tk)

pk−1(tk) = pk(tk) 1 C pk′−1(tk) = pk′(tk)

Cornell CS4620 Fall 2020 Steve Marschner • 7 Deﬁning spline curves

• 2D spline curves are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the knots are at the integers

Cornell CS4620 Fall 2020 Steve Marschner • 8 Deﬁning spline curves

• 2D spline curves are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the knots are at the integers

Cornell CS4620 Fall 2020 Steve Marschner • 8 Deﬁning spline curves

• 2D spline curves are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the knots are at the integers

Cornell CS4620 Fall 2020 Steve Marschner • 8 Deﬁning spline curves

• 2D spline curves are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the knots are at the integers

Cornell CS4620 Fall 2020 Steve Marschner • 8 Deﬁning spline curves

• Generally f(t) is a piecewise polynomial – for this lecture, the discontinuities are at the integers – e.g., a cubic spline has the following form over [k, k + 1):

– Coefﬁcients are different for every interval

Cornell CS4620 Fall 2020 Steve Marschner • 9 Coordinate functions

Cornell CS4620 Fall 2020 Steve Marschner • 10 Coordinate functions

Cornell CS4620 Fall 2020 Steve Marschner • 11 Coordinate functions

Cornell CS4620 Fall 2020 Steve Marschner • 11 Control of spline curves

• Speciﬁed by a sequence of controls (points or vectors) • Shape is guided by control points (aka control polygon) – interpolating: passes through points – approximating: merely guided by points – some splines interpolate only certain points (e.g. endpoints)

Cornell CS4620 Fall 2020 Steve Marschner • 12 Control of spline curves

Cornell CS4620 Fall 2020 Steve Marschner • 12 Spline curves and their controls

• Each coordinate is separate – the function x(t) is determined solely by the x coordinates of the control points – this means 1D, 2D, 3D, … curves are all really the same • Spline curves are linear functions of their controls – moving a control point two inches to the right moves x(t) twice as far as moving it by one inch – x(t), for ﬁxed t, is a linear combination (weighted sum) of the controls’ x coordinates – f(t), for ﬁxed t, is a linear combination (weighted sum) of the controls

Cornell CS4620 Fall 2020 Steve Marschner • 13 Context

• Today we are talking about deﬁning 1D curves, living in any dimension space – emphasizing 2D – higher dimensions are no more complex (just more coords) • Splines can be used to deﬁne objects of any dimension – 2D surfaces – 3D solids – … • Higher dimensions are built from same 1D functions – spline patches have N2 control points – joining patches together is more complicated than curves

Cornell CS4620 Fall 2020 Steve Marschner • 14 Plan 1. Spline segments – how to deﬁne a polynomial on [0,1] – …that has the properties you want – …and is easy to control 2. Spline curves – how to chain together lots of segments – …so that the whole curve has the properties you want – …and is easy to control 3. Reﬁnement and evaluation – how to add detail to splines – how to approximate them with line segments

Cornell CS4620 Fall 2020 Steve Marschner • 15 Spline Segments

Cornell CS4620 Fall 2020 Steve Marschner • 16 A spline segment

• A polynomial function over the interval [tk, tk+1] • When talking about a single segment, to keep things simple, we assume

k = 0; t0 = 0; t1 = 1 – that is, the segment lives on the interval [0,1]

Cornell CS4620 Fall 2020 Steve Marschner • 17 Linear spline (line segment)

• Control points are the vertices

• Each segment will be a linear function p1 = (x1, y1)

– starts at p0 (when t = 0) – ends at p1 (when t = 1) – moves at constant speed along segment – both coordinate functions are linear

p0 = (x0, y0)

Cornell CS4620 Fall 2020 Steve Marschner • 18 Linear interpolation

• Take one coordinate, x • It is linear: x(t) = at + b

– we want x(0) = x0 and x(1) = x1 – this is achieved by b = x0 and a = x1 − x0

x(t) = (x1 − x0)t + x0

Cornell CS4620 Fall 2020 Steve Marschner • 19 Three equivalent notions

x(t) = (x1 − x0)t + x0

= (1 − t)x0 + tx1

= x0b0(t) + x1b1(t)

a linear polynomial with coefﬁcients

that are linear functions of x0 and x1

a linear combination of the values

x0 and x1with weights (1 − t) and t

a linear combination of the functions

b0(t) = 1 − t and b1(t) = t with weights x0 and x1

Cornell CS4620 Fall 2020 Steve Marschner • 20 Spline matrix

• A nice generalizable way of writing this is

−1 1 x0 x(t) = [t 1] [ 1 0] [x1]

Cornell CS4620 Fall 2020 Steve Marschner • 21 Spline matrix

• A nice generalizable way of writing this is

−1 1 x0 x(t) = [t 1] [ 1 0] [x1]

spline monomials controls matrix

Cornell CS4620 Fall 2020 Steve Marschner • 21 Spline matrix

• A nice generalizable way of writing this is

−1 1 x0 x(t) = [t 1] [ 1 0] [x1]

spline monomials controls matrix

−1 1 x0 −1 1 x0 x(t) = [t 1] = [t 1] ([ 1 0] [x1]) ( [ 1 0]) [x1]

group this way to see coefﬁcients group this way to see basis functions

Cornell CS4620 Fall 2020 Steve Marschner • 21 Linear 2D spline segment

• Vector formulation x(t)=(x x )t + x 1 0 0 y(t)=(y y )t + y 1 0 0 f(t)=(p p )t + p 1 0 0 • Matrix formulation

11 p f(t)= t 1 0 10p   1 ⇥ ⇤

Cornell CS4620 Fall 2020 Steve Marschner • 22 Basis function formulation

f(t)

p0 f(t) = b0(t)p0 + b1(t)p1

Cornell CS4620 Fall 2020 Steve Marschner • 23 Hermite splines

• Less trivial example • Form of curve: piecewise cubic • Constraints: endpoints and tangents (derivatives)

Cornell CS4620 Fall 2020 Steve Marschner • 24 Hermite splines in 1D

Cornell CS4620 Fall 2020 Steve Marschner • 25 Hermite splines in 1D

x(t)=at3 + bt2 + ct + d 2 x0(t)=3at +2bt + c

x(0) = x0 = d

x(1) = x1 = a + b + c + d

x0(0) = x0 = c

x0(1) = x10 =3a +2b + c

Cornell CS4620 Fall 2020 Steve Marschner • 25 Hermite splines in 1D

x(t)=at3 + bt2 + ct + d 2 x0(t)=3at +2bt + c

x(0) = x0 = d

x(1) = x1 = a + b + c + d

x0(0) = x0 = c

x0(1) = x10 =3a +2b + c

Cornell CS4620 Fall 2020 Steve Marschner • 25 Hermite splines in 1D

x(t)=at3 + bt2 + ct + d 2 x0(t)=3at +2bt + c

x(0) = x0 = d

x(1) = x1 = a + b + c + d

x0(0) = x0 = c

x0(1) = x10 =3a +2b + c

d = x0

c = x0 a =2x 2x + x0 + x0 0 1 0 1 b = 3x +3x 2x0 x0 0 1 0 1

Cornell CS4620 Fall 2020 Steve Marschner • 25 Hermite splines

• Matrix form is much cleaner

2 21 1 p0 33 2 1 p f(t)= t3 t2 t 1 2 3 2 13 0010 t0 ⇥ ⇤ 6 10007 6t1 7 6 7 6 7 4 5 4 5 – coefﬁcients = rows – basis functions = columns • note the two p columns sum to [0 0 0 1]T

Cornell CS4620 Fall 2020 Steve Marschner • 26 Matrix form of spline

f(t)=at3 + bt2 + ct + d

p0 ⇥⇥⇥⇥ p t3 t2 t 1 1 2⇥⇥⇥⇥3 2p 3 ⇥⇥⇥⇥ 2 ⇥ ⇤ 6 7 6p37 6⇥⇥⇥⇥7 6 7 4 5 4 5

f(t)=b0(t)p0 + b1(t)p1 + b2(t)p2 + b3(t)p3

Cornell CS4620 Fall 2020 Steve Marschner • 27 Matrix form of spline

f(t)=at3 + bt2 + ct + d

p0 ⇥⇥⇥⇥ p t3 t2 t 1 1 2⇥⇥⇥⇥3 2p 3 ⇥⇥⇥⇥ 2 ⇥ ⇤ 6 7 6p37 6⇥⇥⇥⇥7 6 7 4 5 4 5

f(t)=b0(t)p0 + b1(t)p1 + b2(t)p2 + b3(t)p3

Cornell CS4620 Fall 2020 Steve Marschner • 27 Matrix form of spline

f(t)=at3 + bt2 + ct + d

p0 ⇥⇥⇥⇥ p t3 t2 t 1 1 2⇥⇥⇥⇥3 2p 3 ⇥⇥⇥⇥ 2 ⇥ ⇤ 6 7 6p37 6⇥⇥⇥⇥7 6 7 4 5 4 5

f(t)=b0(t)p0 + b1(t)p1 + b2(t)p2 + b3(t)p3

Cornell CS4620 Fall 2020 Steve Marschner • 27 Hermite splines

• Hermite blending functions

b0(t) b1(t)

b2(t)

b3(t)

Cornell CS4620 Fall 2020 Steve Marschner • 28 Hermite to Bézier

• Mixture of points and vectors is awkward • Specify tangents as differences of points

p0 p 1

Cornell CS4620 Fall 2020 Steve Marschner • 29 Hermite to Bézier

• Mixture of points and vectors is awkward • Specify tangents as differences of points

– t1 p0 p 1

Cornell CS4620 Fall 2020 Steve Marschner • 29 Hermite to Bézier

• Mixture of points and vectors is awkward • Specify tangents as differences of points

– t1 p0 p 1

Cornell CS4620 Fall 2020 Steve Marschner • 29 Hermite to Bézier

• Mixture of points and vectors is awkward • Specify tangents as differences of points

q 1 q 2 I’m calling these t 0 points q just for – t this slide and the 1 next one. p0 q 0 q p 3 1

– note derivative is deﬁned as 3 times offset • reason is illustrated by linear case

Cornell CS4620 Fall 2020 Steve Marschner • 29 Hermite to Bézier q 1 q 2 t p0 = q0 0 – t p1 = q3 1 p 0 q t0 = 3(q1 q0) 0 q p 3 1 t = 3(q q ) 1 3 2

p0 1000q0 p 0001q 1 = 1 2v 3 2 33 0 03 2q 3 0 2 6v 7 6 00 337 6q 7 6 17 6 7 6 37 4 5 4 5 4 5

Cornell CS4620 Fall 2020 Steve Marschner • 30 Hermite to Bézier q 1 q 2 t p0 = q0 0 – t p1 = q3 1 p 0 q t0 = 3(q1 q0) 0 q p 3 1 t = 3(q q ) 1 3 2

a 2 21 1 1000q0 b 33 2 1 0001q = 1 2c3 20010 3 2 33 0 03 2q 3 2 6d7 6 10007 6 00 337 6q 7 6 7 6 7 6 7 6 37 4 5 4 5 4 5 4 5

Cornell CS4620 Fall 2020 Steve Marschner • 30 Hermite to Bézier q 1 q 2 t p0 = q0 0 – t p1 = q3 1 p 0 q t0 = 3(q1 q0) 0 q p 3 1 t = 3(q q ) 1 3 2

a 13 31 q0 b 3 630 q = 1 2c3 2 33 003 2q 3 2 6d7 6 10007 6q 7 6 7 6 7 6 37 4 5 4 5 4 5

Cornell CS4620 Fall 2020 Steve Marschner • 30 Bézier matrix

13 31 p0 3 630 p f(t)= t3 t2 t 1 1 2 33 003 2p 3 2 ⇥ ⇤ 6 10007 6p37 6 7 6 7 4 5 4 5 – note that the basis functions are the Bernstein polynomials

n k n k The pk column b (t)= t (1 t) n,k k of the matrix ✓ ◆ deﬁnes the and that deﬁnes Bézier curves for any degree polynomial b3,k(t)

Cornell CS4620 Fall 2020 Steve Marschner • 31 Bézier basis

Cornell CS4620 Fall 2020 Steve Marschner • 32 Another way to Bézier segments

• A really boring spline segment: f(t) = p0 – it only has one control point – the curve stays at that point for the whole time • Only good for building a piecewise constant spline – a.k.a. a set of points

Cornell CS4620 Fall 2020 Steve Marschner • 33 Another way to Bézier segments

• A piecewise linear spline segment – two control points per segment p – blend them with weights α and β = 1 – α 1 • Good for building a piecewise linear spline – a.k.a. a polygon or polyline

p0 + p1

p0 Cornell CS4620 Fall 2020 Steve Marschner • 34 Another way to Bézier segments

These labels show p0 + p1 the weights, not the distances.

p0 Cornell CS4620 Fall 2020 Steve Marschner • 34 Another way to Bézier segments

• A linear blend of two piecewise linear segments – three control points now – interpolate on both segments using α and β – blend the results with the same weights • makes a quadratic spline segment – ﬁnally, a curve! p1,0 = ↵p0 + p1

p1,1 = ↵p1 + p2

p2,0 = ↵p1,0 + p1,1

= ↵↵p0 + ↵p1 + ↵p1 + p2 2 2 = ↵ p0 +2↵p1 + p2

Cornell CS4620 Fall 2020 Steve Marschner • 35 p1

2 2 p0 + 2p1 + p1

Cornell CS4620 Fall 2020 Steve Marschner • 36 Another way to Bézier segments

• Cubic segment: blend of two quadratic segments – four control points now (overlapping sets of 3) – interpolate on each quadratic using α and β – blend the results with the same weights • makes a cubic spline segment – this is the familiar one for graphics—but you can keep going

p3,0 =↵p2,0 + p2,1

=↵↵↵p0 + ↵↵p1 + ↵↵p1 + ↵p2

↵↵p1 + ↵p2 + ↵p2 + p3 3 2 2 3 =↵ p0 +3↵ p1 +3↵ p2 + p3

Cornell CS4620 Fall 2020 Steve Marschner • 37 p1 p2

3 2 2 3 p0 + 3 p1 + 3 p2 + p3

Cornell CS4620 Fall 2020 Steve Marschner • 38 de Casteljau’s algorithm

• A recurrence for computing points on Bézier spline segments:

p0,i = pi

pn,i = ↵pn 1,i + pn 1,i+1

• Cool additional feature: also subdivides the segment into two shorter ones [FvDFH]

Cornell CS4620 Fall 2020 Steve Marschner • 39 Cubic Bézier splines

• Very widely used type, especially in 2D – e.g. it is a primitive in PostScript/PDF • Can represent smooth curves with corners • Nice de Casteljau recurrence for evaluation • Can easily add points at any position • Illustrator demo

Cornell CS4620 Fall 2020 Steve Marschner • 40 Spline segment properties

• Convex hull property – convex hull = smallest convex region containing points • think of a rubber band around some pins – some splines stay inside convex hull of control points • make clipping, culling, picking, etc. simpler

YES YES YES NO

Cornell CS4620 Fall 2020 Steve Marschner • 41 Convex hull

• If basis functions are all positive, the spline has the convex hull property – we’re still requiring them to sum to 1

– if any basis function is ever negative, no convex hull prop. • proof: take the other three points at the same place

Cornell CS4620 Fall 2020 Steve Marschner • 42 Afﬁne invariance

• Transforming the control points is the same as transforming the curve – true for all commonly used splines – extremely convenient in practice…

Cornell CS4620 Fall 2020 Steve Marschner • 43 Afﬁne invariance

• Transforming the control points is the same as transforming the curve – true for all commonly used splines – extremely convenient in practice…

Cornell CS4620 Fall 2020 Steve Marschner • 43 Afﬁne invariance

• Transforming the control points is the same as transforming the curve – true for all commonly used splines – extremely convenient in practice…

Cornell CS4620 Fall 2020 Steve Marschner • 43 Spline Curves

Cornell CS4620 Fall 2020 Steve Marschner • 44 Chaining spline segments

• Can only do so much with a single polynomial • Can use these functions as segments of a longer curve – curve from t = 0 to t = 1 defined by first segment – curve from t = 1 to t = 2 defined by second segment

f(t)=f (t i) for i t i +1 i   • To avoid discontinuity, match derivatives at junctions – this produces a C1 curve

Cornell CS4620 Fall 2020 Steve Marschner • 45 Continuity

• Smoothness can be described by degree of continuity – zero-order (C0): position matches from both sides – ﬁrst-order (C1): tangent matches from both sides – second-order (C2): curvature matches from both sides – Gn vs. Cn

zero order ﬁrst order second order

Cornell CS4620 Fall 2020 Steve Marschner • 46 Continuity

• Parametric continuity (C) of spline is continuity of coordinate functions • Geometric continuity (G) is continuity of the curve itself • Neither form of continuity is guaranteed by the other – Can be C1 but not G1 when p(t) comes to a halt (next slide) – Can be G1 but not C1 when the tangent vector changes length abruptly

Cornell CS4620 Fall 2020 Steve Marschner • 47 Geometric vs. parametric continuity

Cornell CS4620 Fall 2020 Steve Marschner • 48 Geometric vs. parametric continuity

Cornell CS4620 Fall 2020 Steve Marschner • 48 Control

• Local control – changing control point only affects a limited part of spline – without this, splines are very difﬁcult to use – many likely formulations lack this • natural spline • polynomial ﬁts

Cornell CS4620 Fall 2020 Steve Marschner • 49 Trivial example: piecewise linear

• Basis function formulation: “function times point” – basis functions: contribution of each point as t changes

– can think of them as blending functions glued together – this is just like a reconstruction ﬁlter!

Cornell CS4620 Fall 2020 Steve Marschner • 50 Trivial example: piecewise linear

• Basis function formulation: “function times point” – basis functions: contribution of each point as t changes

– can think of them as blending functions glued together – this is just like a reconstruction ﬁlter!

Cornell CS4620 Fall 2020 Steve Marschner • 50 Splines as reconstruction

Cornell CS4620 Fall 2020 Steve Marschner • 51 Splines as reconstruction

Cornell CS4620 Fall 2020 Steve Marschner • 51 Seeing the basis functions

• Basis functions of a spline are revealed by how the curve changes in response to a change in one control – to get a graph of the basis function, start with the curve laid out in a straight, constant-speed line • what are x(t) and y(t)? – then move one control straight up

Cornell CS4620 Fall 2020 Steve Marschner • 52 Seeing the basis functions

Cornell CS4620 Fall 2020 Steve Marschner • 52 Hermite splines

• Controls are endpoints ti and endpoint tangents

• Segments are chained pi

by sharing points and pi+1 tangents between

adjacent segments ti+1

2 21 2 p0 33 2 1 p f(t)= t3 t2 t 1 2 3 2 13 0010p0 ⇥ ⇤ 6 10007 6p0 7 6 7 6 17 4 5 4 5 Cornell CS4620 Fall 2020 Steve Marschner • 53 Hermite basis

Cornell CS4620 Fall 2020 Steve Marschner • 54 t Hermite basis i

pi+1

ti+1

1 p p 0 1

t0 0 0 1 t1

Cornell CS4620 Fall 2020 Steve Marschner • 54 t Hermite basis i

pi+1 pi–1 t t i+1 i–1 t i+2 pi+2

pi pi+1

t i+1 0 i – 1 i i + 1 i + 2 ti

Cornell CS4620 Fall 2020 Steve Marschner • 54 Bézier basis

Cornell CS4620 Fall 2020 Steve Marschner • 55 Chaining Bézier splines

• No continuity built in • Achieve C1 using collinear control points

Cornell CS4620 Fall 2020 Steve Marschner • 56 Chaining Bézier splines

• No continuity built in • Achieve C1 using collinear control points

Cornell CS4620 Fall 2020 Steve Marschner • 56 Making long uniform splines

• Hermite curves are convenient because they can be made long easily • Bézier curves are convenient because their controls are all points – but it is fussy to maintain continuity constraints – and they interpolate every 3rd point, which is a little odd • We derived Bézier from Hermite by deﬁning tangents from control points – a similar construction leads to the interpolating Catmull-Rom spline

Cornell CS4620 Fall 2020 Steve Marschner • 57 Hermite to Catmull-Rom