2D Spline Curves CS 4620 Lecture 18 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 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 2013 • Lecture 16 © 2013 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 2013 • Lecture 16 © 2013 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 2013 • Lecture 16 © 2013 Steve Marschner • 4 Defining spline curves • At the most general they are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the discontinuities are at the integers Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 5 Defining spline curves • At the most general they are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the discontinuities are at the integers Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 5 Defining spline curves • At the most general they are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the discontinuities are at the integers Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 5 Defining spline curves • At the most general they are parametric curves S = f(t) t [0,N] { | 2 } • For splines, f(t) is piecewise polynomial – for this lecture, the discontinuities are at the integers Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 5 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 2013 • Lecture 16 © 2013 Steve Marschner • 6 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 7 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 7 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 7 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 7 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 8 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 8 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 8 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 8 Coordinate functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 8 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 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 9 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 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 9 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 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 9 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 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 9 How splines depend on 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 2013 • Lecture 16 © 2013 Steve Marschner • 10 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 2013 • Lecture 16 © 2013 Steve Marschner • 11 Spline Segments Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 12 Trivial example: piecewise linear • This spline is just a polygon – control points are the vertices • But we can derive it anyway as an illustration • Each interval will be a linear function – x(t) = at + b – constraints are values at endpoints – b = x0 ; a = x1 – x0 – this is linear interpolation Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 13 Trivial example: piecewise linear • 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 2013 • Lecture 16 © 2013 Steve Marschner • 14 Trivial example: piecewise linear • Basis function formulation – regroup expression by p rather than t f(t)=(p p )t + p 1 − 0 0 =(1 t)p + tp − 0 1 – interpretation in matrix viewpoint 11 p f(t)= t 1 0 −10 p ✓ ◆ 1 ⇥ ⇤ Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 15 Trivial example: piecewise linear • Vector blending formulation: “average of points” – blending functions: contribution of each point as t changes Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 16 Hermite splines • Less trivial example • Form of curve: piecewise cubic • Constraints: endpoints and tangents (derivatives) t0 p0 p1 t1 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 17 Hermite splines • Solve constraints to find coefficients x(t)=at3 + bt2 + ct + d 2 x0(t)=3at +2bt + c d = x0 x(0) = x0 = d c = x0 x(1) = x1 = a + b + c + d a =2x 2x + x0 + x0 0 − 1 0 1 x0(0) = x0 = c b = 3x +3x 2x0 x0 0 − 0 1 − 0 − 1 x0(1) = x10 =3a +2b + c Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 18 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 2013 • Lecture 16 © 2013 Steve Marschner • 19 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 2013 • Lecture 16 © 2013 Steve Marschner • 19 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 2013 • Lecture 16 © 2013 Steve Marschner • 19 Hermite splines • Matrix form is much simpler 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 p columns sum to [0 0 0 1]T Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 20 Hermite splines • Hermite blending functions Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 21 Hermite to Bézier • Mixture of points and vectors is awkward • Specify tangents as differences of points t 0 p0 p1 t1 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 22 Hermite to Bézier • Mixture of points and vectors is awkward • Specify tangents as differences of points t 0 – t1 p0 p1 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 22 Hermite to Bézier • Mixture of points and vectors is awkward • Specify tangents as differences of points t 0 – t1 p0 p1 Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 22 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 q0 q 3 p1 – note derivative is defined as 3 times offset • reason is illustrated by linear case Cornell CS4620 Fall 2013 • Lecture 16 © 2013 Steve Marschner • 22 Hermite to Bézier q1 q2 t p0 = q0 0 – t p1 = q3 1 p0 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 2013 • Lecture 16 © 2013 Steve Marschner • 23 Hermite to Bézier q1 q2 t p0 = q0 0 – t p1 = q3 1 p0 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 2−0010− − 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 2013 • Lecture