CS 450: Computer Graphics PARAMETRIC SPLINES AND SURFACES
Spring 2015 Dr. Michael J. Reale INTRODUCTION Mostly, weve concentrated on objects that are made up of triangles Hardware is set up to render triangles Objects and animation paths can have many different underlying geometric descriptions Example: curves and curved surfaces can be described precisely by equations equations can then beevaluated, and triangles can be created and rendered ADVANTAGES OF CURVES AnD CURVED SURFACES
1) Compact representation E.g., can define a curve precisely with 4 control points (instead of approximating it with a bunch of lines) 2) Scalable I.e., can turn into 2 polygons or 2000 polygons, depending on how much detail you want 3) Smoother and more continuous primitives 4) Animation and collision detection may be simpler and faster I.e., using underlying mathematical descriptions rather than checking each polygon with every other polygon Bonus advantages for real-time rendering: 1) Less memory needed for representation 2) Fewer points to transform 3) If graphics hardware DOES handle curved surface descriptions directly, less to transfer from CPU to GPU OVERVIEW We will touch on the following subjects:
Parametric Curves Parametric Curved Surfaces Although there are many kinds of curves/splines/patches out there, we will focus on Bzier curves andBzier patches PARAMETRIC CURVES INTRODUCTION Parametric curve = curve that is a function of a parameter t p(t) t = 0 start of the curve t = 1 end of the curve Increments of t evenly-spaced distances along curve path great for animations or camera paths! When dealing with 2D or 3D points, each dimension is a separate function of t x(t), y(t), z(t) For the slides that follow, however, we will look at a single function p(t) that outputs a point (whatever itsdimension) LINEAR INTERPOLATION Linear interpolation = traces out straight line:
p0 = starting point, p1 = ending point Problem: more than two points in your path sudden jerks as you jump from one line segment to thenext (at joints) discontinuous at joints There must be a better way What we get with linear interpolation What we want INTRODUCING BEZIER CURVES
Bzier curves (pronounced beh-ze-eh) Basic idea is repeatedly (and recursively) doing linear interpolation called de Casteljau algorithm Lets say we have three control points: a, b, and c To get a point along the curve (say, at t = 1/3) Get d = 1/3 of the way from a to b Get e = 1/3 of the way from b to c Get f = p(1/3) = 1/3 of the way from d to e In general: BEZIER CURVE DEGREES AND CONTROL POINTS
The maximum degree of this example is n = 2 (quadratic) The number of control points is (n+1) = 3 Given a Bezier curve of degree n, the number of control points is (n+1) BEZIER CURVES: DE CASTELJAU ALGORITHM
A given point on the curve is defined recursively Control points: p1 , p2 , p3 , After applying linear interpolation k times intermediate control points pik are obtained Previous example: BEZIER CURVES: DEFINITION
A Bezier curve with (n+1) initial control points can be described recursively: A single point on the curve is defined by: BEZIER CURVES: PREVIOUS EXAMPLE
So, how does this work?Lets look at our previous example We already know our initial control points (k = 0): Get the next interpolation level (k = 1): Final level (k = 2 = n): BEZIER CURVES: ANOTHER EXAMPLE
Lets say we have a quartic Bezier curve Degree n = 4 Number of control points (n+1) = 5 We know values of initial control points Work our way up tree, interpolating as we go BEZIER CURVES USING BERNSTEIN POLYNOMIALS
Remember how earlier we had turned our quadratic Bezier curve into a nice tidy algebraic formula? Turns out we can do the same with EVERY Bezier curve WITHOUT having to do the repeatedinterpolation use Bernstein form: This form uses Bernstein polynomials (first term is binomial coefficient): BEZIER CURVES USING BERNSTEIN POLYNOMIALS
Two properties of Bernstein polynomials: Will stay between 0 and 1 if t is between 0 and 1 All polynomials sum to 1 Means curve will stay close to control points In fact, a Bezier curve fits within convex hull of control points: BEZIER CURVES USING BERNSTEIN POLYNOMIALS
Bernstein polynomials also called blending functions or Bezier basis functions n = 1 n = 2 n = 3 BEZIER CURVES in MATRIX FORM
It is comparatively easy to define a Bezier curve with a matrix Create a matrix that contains the correct Bernstein polynomial values Create another matrix for the control points Reuse both of these matrices for each value of t you need Example: a cubic Bezier curve in matrix form DISADVANTAGES OF BEZIER CURVES
Do not pass through all control points (except endpoints) Degree increases with number of control points more expensive to evaluate One solution to the latter problem: piecewise Bezier curves PIECEWISE BEZIER CURVES
Lets say we have two cubic Bezier curves: q0 , q1 , q2 , q3 r0 , r1 , r2 , r3 We want to join these such that: p(0) = q0 p(1) = q3 = r0 p(3) = r3 How do we do this? PIECEWISE BEZIER CURVES
First of all, we have a t value that can be greater than 1: If t is [0,1] use first Bezier curve If t > 1 scale and translate value and use t on second Bezier curve: PIECEWISE BEZIER CURVES: C0 CONTINUITY
To connect the curves, we could just set q3 = r0 HOWEVER, there could be a sudden change indirection at joint derivative not continuous This gives us C0 continuity no gaps in position PIECEWISE BEZIER CURVES: G1 CONTINUITY
Tangent of curve at q0 parallel with (q1 q0 ) Tangent of curve at q3 parallel with (q3 q2 ) So, make sure (incoming tangent at joint) = (outgoing tangent at joint): This gives us G1 continuity Says nothing about the lengths of the tangents Gn = geometric continuity weaker than corresponding Cn continuity PIECEWISE BEZIER CURVES: C1 CONTINUITY
In previous example, derivatives would be continuous if c = 1.0 implies t2 = 2.0 HOWEVER, we want t2 = 3.0 To get even better continuity, we can define c as follows: This gives us C1 continuity derivative is continuous (even at joints) I.e., tangent vectors point in same direction and are of same length C0 continuity (position)
G1 continuity (tangent directions) C1 continuity (derivatives) PARAMETRIC SURFACES INTRODUCTION Parametric surfaces = natural extension of parametric curves Similar to the jump from lines to triangles/polygons Often tessellated by evaluating surface at difference positions and then building triangles for rendering Patch = surface with rectangular domain Here we will discuss Bezier patches BEZIER PATCHES Bezier patches = extends Bezier curves by:
Using two parameters (u,v) Using bilinear interpolation Also lies in convex hull of control points Start with four points (instead of two): a, b, c, d Interpolate a & b and c & d to get e and f: Finally, linearly interpolate between e & f using v: BEZIER PATCHES More points can be added (i.e., degree can be increased) Example: biquadratic Bezier surface with 9 control points (n = 2, # of control points = (n+1)*(n+1) = 3*3) BEZIER PATCHES IN BERNSTEIN FORM
In Bernstein form, a Bezier patch is defined by: m and n = degree of the surface Compound degree = m x n BEZIER PATCHES IN BERNSTEIN FORM: ANOTHER VIEW
If we reinterpret the previous equation We suddenly notice that qi(v) is a Bezier curve so, basically, weare computing the points along each Bezier curve with a fixed vvalue, then computing the value from a Bezier curve made fromTHOSE points BEZIER PATCH: EXAMPLE Generated quads 4 x 4 control points
Shaded quads Move one control point Most change near control point PIECEWISE BEZIER PATCHES
Usually you will want to stitch patches together to get a complete surface However, C0 continuity is going to look terrible Can enforce C1 continuity by doing the following: Take the two rows near boundary from each patch: a2j and b1j For each j a2j , b0j , and b1j : Must be collinear (i.e., lie on line) Must have same ratio (with k constant for all j): PIECEWISE BEZIER PATCHES
Problem with previous approach: lose a lot of degrees of freedom (dont have as much control overpatches) Even worse at corner where 4 patches meet 9 points all in same plane and evenly spaced If you only want G1 continuity at corners (and only there), can relax to only make them coplanar PIECEWISE BEZIER PATCHES
C1 continuity required for good-looking texturing G1 continuity reasonable for reflections and shading, but C1 continuity is better C0 continuity (position) C1 continuity (derivative)