Computer Graphics Si Lu Fall 2017 - PowerPoint PPT Presentation

Computer Graphics Si Lu Fall 2017 http://web.cecs.pdx.edu/~lusi/CS447/CS447_547_Comp uter_Graphics.htm 11/20/2017

Last time o Modeling Techniques 2

Today o More modeling techniques o Splines o Homework 5 available, due 11/29 in class o Pick a slot for grading Project 2 n https://docs.google.com/spreadsheets/d/1wiiB91GLX 3dkZMARjPR- PIA2qcI7lgcGGyzCAvMLZPY/edit#gid=0 3

Implicit Functions o Some surfaces can be represented as the vanishing points of functions (defined over 3D space) Places where a function f(x,y,z)=0 n o Some objects are easy to represent this way n Spheres, ellipses, and similar n More generally, quadratic surfaces:        2 2 2 ax bx cy dy ez fz g 0 n Shapes depends on all the parameters a,b,c,d,e,f,g

Blobs and Metaballs o Define the location of some points, p i o For each point, define a function on the distance to a given point, D (x,p i ) o Sum these functions up, and use them as an implicit function o Often, use Gaussian functions of distance, or other forms Various results are called blobs or metaballs n

Example with Blobs Rendered with POVray. Not everything is a blob, but the characters are.

Blob Math n blobs  o Implicit equation:    f ( x , y , z ) w w g ( x , y , z ) 0 0 i i  i 1 o The w i are weights – just numbers o The g i are functions, one common choice is: 2   x c i  g i ( x ) e  i c i and  i are parameters n

Rendering Implicit Surfaces o Some methods can render them directly n Raytracing - find intersections with Newton’s method o For polygonal renderer, must convert to polygons n Marching Cubes algorithm n Also used for finding iso-surfaces , or level sets , in volume data

Implicit Surfaces Summary o Advantages: n Good for organic looking shapes eg human body n Good for extracting surfaces from volume representations, such as water surfaces in fluid simulation n Easy inside/outside testing o Disadvantages: n Difficult to render Difficult to control when animating n

What are Parametric Curves? o Define a parameter space 1D for curves: t n 2D for surfaces: ( s , t ) n o Define a mapping from parameter space to 3D points n A function that takes parameter values and gives back 3D points o The result is a parametric curve or surface ( F x ( t ), F y ( t )) Mapping: F:t → (x,y) 1 0 1 t 0

Why Parametric Curves? o Parametric curves are intended to provide the generality of polygon meshes but with fewer parameters for smooth surfaces n Polygon meshes have as many parameters as there are vertices (at least) o Fewer parameters make it faster to create a curve, and easier to edit an existing curve o Normal vectors and texture coordinates can be easily defined everywhere o Parametric curves are easier to animate than polygon meshes

Parametric Curves o We have seen the parametric form for a line:    x ( 1 t ) x tx 0 1    y ( 1 t ) y ty 0 1    z ( 1 t ) z tz 0 1 o Note that x, y and z are each given by an equation that involves: The parameter t n Some user specified control points, x 0 and x 1 n o This is an example of a parametric curve

Basis Functions (first sighting) o A line is the sum of two functions multiplied by vectors:       x x x 0 1            y 1 t y t y       0 1       z z z       0 1 o A linear combination of basis functions n t and 1- t are the basis functions o The weights are called control points (x 0 ,y 0 ,z 0 ) and (x 1 ,y 1 ,z 1 ) are the control points n n They control the shape and position of the curve

Hermite Spline o A spline is a parametric curve defined by control points n The term spline dates from engineering drawing, where a spline was a piece of flexible wood used to draw smooth curves The control points are adjusted by the user to control the n shape of the curve o A Hermite spline is a curve for which the user provides: n The endpoints of the curve The parametric derivatives of the curve at the endpoints n (tangents with length) o The parametric derivatives are dx/dt, dy/dt, dz/dt n That is enough to define a cubic Hermite spline

Spline 15 Source: http://en.wikipedia.org/wiki/File:Spline_(PSF).png

Control Point Interpretation End Tangent Start Tangent d x d x dt dt 1 0 x 1 End Point x 0 Start Point

Hermite Spline (2) d x d x o Say the user provides x , x , 0 , 1 0 1 dt dt 0 1 o A cubic spline has degree 3, and is of the form:     3 2 x at bt ct d For some constants a, b, c and d derived from the control n points, but how? o We have constraints: The curve must pass through x 0 when t=0 n The derivative must be x’ 0 when t=0 n n The curve must pass through x 1 when t=1 n The derivative must be x’ 1 when t=1

Hermite Spline (3)        a 2 x 2 x x x o Solving for the unknowns gives: 1 0 1 0       b 3 x 3 x x 2 x 1 0 1 0   c x 0  d x 0 o Rearranging gives:     2 3 0 0  3 t    3 2 x x ( 2 t 3 t )     1  2 3 0 1 2 t      3 2 x ( 2 t 3 t 1 )       or  x x x x x 0   1 0 1 0       1 1 0 0 3 2 x ( t t ) t 1         3 2 x ( t 2 t t )  1 2 1 0 1     0

Basis Functions o A point on a Hermite curve is obtained by multiplying each control point by some function and summing

Splines in 2D and 3D o For higher dimensions, define the control points in higher dimensions (that is, as vectors)      2 3 0 0 3 t       x x x x x     1 0 1 0  2 3 0 1 2 t            y y y y y         1 0 1 0  1 1 0 0 t         z z z z z       1 0 1 0  1 2 1 0   1    

Bezier Curves (1) o Different choices of basis functions give different curves Choice of basis determines how the control points influence n the curve n In Hermite case, two control points define endpoints, and two more define parametric derivatives o For Bezier curves, two control points define endpoints, and two control the tangents at the endpoints in a geometric way

Control Point Interpretation Point along start tangent x 1 x 3 End Point x x 2 0 Point along end Tangent Start Point

Bezier Curves (2) o The user supplies d +1 control points, p i o Write the curve as:   d d           d i     d i  B t t 1 t d x t p B t   i i i i    i 0 o The functions B id are the Bernstein polynomials of degree d o This equation can be written as a matrix equation also n There is a matrix to take Hermite control points to Bezier control points

Bezier Basis Functions for d =3

Bezier Curves of Varying Degree

Bezier Curve Properties o The first and last control points are interpolated o The tangent to the curve at the first control point is along the line joining the first and second control points o The tangent at the last control point is along the line joining the second last and last control points o The curve lies entirely within the convex hull of its control points The Bernstein polynomials (the basis functions) sum to 1 and n are everywhere positive o They can be rendered in many ways n E.g.: Convert to line segments with a subdivision algorithm

Rendering Bezier Curves (1) o Evaluate the curve at a fixed set of parameter values and join the points with straight lines o Advantage: Very simple o Disadvantages: n Expensive to evaluate the curve at many points n No easy way of knowing how fine to sample points, and maybe sampling rate must be different along curve n No easy way to adapt. In particular, it is hard to measure the deviation of a line segment from the exact curve

Rendering Bezier Curves (2) o Recall that a Bezier curve lies entirely within the convex hull of its control vertices o If the control vertices are nearly collinear, then the convex hull is a good approximation to the curve o Also, a cubic Bezier curve can be subdivided into two shorter curves that exactly cover the original o This suggests an algorithm: Keep breaking the curve into sub-curves n Stop when the control points of each sub-curve are nearly n collinear n Draw the control polygon - the polygon formed by the control points