Curve & Surface.

Slides:



Advertisements
Similar presentations
Curves Jim Van Verth Essential Math for Games Animation Problem: want to replay stored set of transformations  Generated by.
Advertisements

Lecture Notes #11 Curves and Surfaces II
© University of Wisconsin, CS559 Spring 2004
2002 by Jim X. Chen: Bezier Curve Bezier Curve.
Lecture 10 Curves and Surfaces I
Geometric Modeling Notes on Curve and Surface Continuity Parts of Mortenson, Farin, Angel, Hill and others.
© University of Wisconsin, CS559 Spring 2004
B-Spline Blending Functions
Dr. S.M. Malaek Assistant: M. Younesi
1 Curves and Surfaces. 2 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized.
Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 12: Spline Curves (review) Ravi Ramamoorthi Most material.
Modeling of curves Needs a ways of representing curves: Reproducible - the representation should give the same curve every time; Computationally Quick;
1. 2 The use of curved surfaces allows for a higher level of modeling, especially for the construction of highly realistic models. There are several approaches.
Geometric Modeling Surfaces Mortenson Chapter 6 and Angel Chapter 9.
Curves Mortenson Chapter 2-5 and Angel Chapter 9
ENDS 375 Foundations of Visualization Geometric Representation 10/5/04.
Modelling: Curves Week 11, Wed Mar 23
University of British Columbia CPSC 414 Computer Graphics © Tamara Munzner 1 Curves Week 13, Mon 24 Nov 2003.
COEN Computer Graphics I
1 Representing Curves and Surfaces. 2 Introduction We need smooth curves and surfaces in many applications: –model real world objects –computer-aided.
Curve Surfaces June 4, Examples of Curve Surfaces Spheres The body of a car Almost everything in nature.
Computer Graphics Lecture 13 Curves and Surfaces I.
Curves and Surfaces (cont’) Amy Zhang. Conversion between Representations  Example: Convert a curve from a cubic B-spline curve to the Bézier form:
Curve Modeling Bézier Curves
Curves and Surfaces CSE3AGR - Paul Taylor Polynomials of Degree n Degree is equal to the highest exponent of a term. Higher exponents result in.
11/19/02 (c) 2002, University of Wisconsin, CS 559 Last Time Many, many modeling techniques –Polygon meshes –Parametric instancing –Hierarchical modeling.
A D V A N C E D C O M P U T E R G R A P H I C S CMSC 635 January 15, 2013 Spline curves 1/23 Curves and Surfaces.
CS 376 Introduction to Computer Graphics 04 / 23 / 2007 Instructor: Michael Eckmann.
V. Space Curves Types of curves Explicit Implicit Parametric.
Quadratic Surfaces. SPLINE REPRESENTATIONS a spline is a flexible strip used to produce a smooth curve through a designated set of points. We.
CS 376 Introduction to Computer Graphics 04 / 20 / 2007 Instructor: Michael Eckmann.
Chapter VI Parametric Curves and Surfaces
INTERPOLATION & APPROXIMATION. Curve algorithm General curve shape may be generated using method of –Interpolation (also known as curve fitting) Curve.
Parametric Surfaces Define points on the surface in terms of two parameters Simplest case: bilinear interpolation s t s x(s,t)x(s,t) P 0,0 P 1,0 P 1,1.
COLLEGE OF ENGINEERING UNIVERSITY OF PORTO COMPUTER GRAPHICS AND INTERFACES / GRAPHICS SYSTEMS JGB / AAS Representation of Curves and Surfaces Graphics.
CS 376 Introduction to Computer Graphics 04 / 25 / 2007 Instructor: Michael Eckmann.
Basic Theory (for curve 02). 1.3 Parametric Curves  The main aim of computer graphics is to display an arbitrary surface so that it looks real.  The.
Representation of Curves & Surfaces Prof. Lizhuang Ma Shanghai Jiao Tong University.
04/18/02(c) 2002 University of Wisconsin Last Time Hermite Curves Bezier Curves.
Curves: ch 4 of McConnell General problem with constructing curves: how to create curves that are “smooth” CAD problem Curves could be composed of segments.
Parametric Curves & Surfaces
Greg Humphreys CS445: Intro Graphics University of Virginia, Fall 2003 Parametric Curves & Surfaces Greg Humphreys University of Virginia CS 445, Spring.
Computing & Information Sciences Kansas State University Lecture 31 of 42CIS 636/736: (Introduction to) Computer Graphics Lecture 32 of 42 Wednesday, 11.
11/26/02(C) University of Wisconsin Last Time BSplines.
Parametric Curves CS 318 Interactive Computer Graphics John C. Hart.
(c) 2002 University of Wisconsin
1 Graphics CSCI 343, Fall 2015 Lecture 34 Curves and Surfaces III.
CS 450: Computer Graphics PARAMETRIC SPLINES AND SURFACES
CS 325 Computer Graphics 04 / 30 / 2010 Instructor: Michael Eckmann.
Rendering Bezier Curves (1) Evaluate the curve at a fixed set of parameter values and join the points with straight lines Advantage: Very simple Disadvantages:
1 대상물체의 형상화를 위해 사용되는 기술  인공물체 : 기하학적 Primitive ( 선, 면, 구, 육면체 ) 등을 이 용하여 형상화. 입력물 : 형상화 물체의 3 차원 좌표값 출력물 : 선구조형상 (Wire framed objects) Technique:Geometric.
Curves University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2013 Tamara Munzner.
Object Modeling: Curves and Surfaces CEng 477 Introduction to Computer Graphics.
Introduction to Parametric Curve and Surface Modeling.
© University of Wisconsin, CS559 Spring 2004
Introduction to Parametric Curve and Surface Modeling
CS5500 Computer Graphics May 11, 2006
Representation of Curves & Surfaces
Chapter 10-2: Curves.
Advanced Computer Graphics: Parametric Curves and Surfaces
© University of Wisconsin, CS559 Fall 2004
© University of Wisconsin, CS559 Fall 2004
Rendering Curves and Surfaces
CURVES CAD/CAM/CAE.
© University of Wisconsin, CS559 Spring 2004
Chapter XVII Parametric Curves and Surfaces
Implicit Functions Some surfaces can be represented as the vanishing points of functions (defined over 3D space) Places where a function f(x,y,z)=0 Some.
UNIT-5 Curves and Surfaces.
Three-Dimensional Object Representation
Introduction to Parametric Curve and Surface Modeling
Presentation transcript:

Curve & Surface

Three basic forms of curves (1/4) Explicit form y = f(x) impossible to get multiple values for a single x break curves like circles and ellipses into segments not invariant with rotation rotation might require further segment breaking problem with curves with vertical tangents infinite slope is difficult to represent Implicit form f(x, y, z) = 0 equation may have more solutions than we want circle: x² + y² = 1, half circle: ? problem to join curve segments together difficult to determine if their tangent directions agree at their joint point So, we use parametric form for curves and surfaces

Three basic forms of curves (2/4) Parametric form : x = x(t), y = y(t), z = z(t) overcomes problems with explicit and implicit forms no geometric slopes (which may be infinite) parametric tangent vectors instead (never infinite) a curve is approximated by a piecewise polynomial curve

Three basic forms of curves (3/4) Parametric form : based on the curve length For parameter t, we obtain equation we denote x=x(t), y=y(t) position vector p(t)=[x(t), y(t)] derivative slope of curve = axis independent because point on a curve is specified by a single value of parameter t. end point and length are fixed by parameter range usually normalized to

Three basic forms of curves (4/4) Ex) parametric rep. Of straight line from position vector to - each components of P(t) has a parametric representation when slope

Parametric Curve (1/6) Three Major types of Curves Hermite Bezier Two end points & two end point tangent vectors Bezier Defined by two end points and two other points that control the end point tangent vectors Spline several kinds, each defined by four points uniform B-splines, non-uniform B-splines, ß-splines

Parametric Curve (2/6) Hermite Curve Bezier Curve Two end points & two end point tangent vectors Bezier Curve two end points and two other points that control the end point tangent vectors

Parametric Curve (3/6) Bezier curve properties The first and last control points are the end points of the curve segment Convexhull property The curve is contained in shaded area formed from the control points The control points do not exert ‘local’ control Moving any control point affect all of the curve to a greater or lesser extent

Parametric Curve (4/6) B-spline curve property four control points convex hull property The curve is transformed by applying affine transformation A B-spline curve exhibit local control

Parametric Curve (5/6) Geometric continuity Parametric continuity G0 geometric continuity two curve segment join together G1 geometric continuity the directions of the two segment’s tangent vectors are equal at a join point not necessarily the magnitudes Parametric continuity C1parametric continuity tangent vectors of two curve segments are equal at the join point direction and magnitude Cn parametric continuity the direction and magnitude of dn/dtn[Q(t)] through the nth derivative are equal at the join point

Parametric Curve (6/6) Subdivision Purpose : To render a curve, rather than evaluate points along a curve, it is more cheaper to subdivide recursively until the convex hulls are sufficiently good approximation to the curve. The termination criterion. linearity test applied to the convex hull. i.e., how far the interior control points deviate from the line connecting the two outer control points.

Biparametric patch (1/3) Each patch is defined by blending control points

Biparametric patch (2/3)

Biparametric patch (3/3) Joining Two patch C0 continuity requires aligning boundary curves C1 continuity requires aligning boundary curves and derivatives

Parametric Surfaces Advantages: Disadvantages: Easy to enumerate points on surface Possible to describe complex shapes Disadvantages: Control mesh must be quadrilaterals Continuity constraints difficult to maintain: C0 easy, C1 possible, C2 hard at extraordinary vertices Hard to find intersections