UCSD CSE 167 - Curved Surfaces - D339733

Home> Schools> University of California, San Diego> Computer Science & Engineering (CSE) > CSE 167> Curved Surfaces

UCSD CSE 167 - Curved Surfaces

School name University of California, San Diego

Pages 48

Download Save

Unformatted text preview:

Curved SurfacesBezier CurvesCurve Evaluationde Casteljau AlgorithmSlide 5Slide 6Slide 7Bezier CurveRecursive Linear InterpolationExpanding the LerpsBernstein PolynomialsSlide 12Cubic Equation FormCubic Matrix FormSlide 15Matrix FormBezier Curves & Cubic CurvesDerivativesTangentsConvex Hull PropertyContinuityInterpolation / ApproximationPiecewise CurvesConnecting Bezier CurvesBezier SurfacesSlide 26Control MeshSurface EvaluationSlide 29Slide 30Slide 31Slide 32Slide 33Slide 34NormalsBezier Surface PropertiesTessellationUniform TessellationAdaptive TessellationMixed TessellationDisplacement MappingScan ConversionTransparencyScanline RenderingSlide 45Surface RenderersSlide 47Other Curve TypesCurved SurfacesCSE167: Computer GraphicsInstructor: Steve RotenbergUCSD, Fall 2005Bezier CurvesBezier curves can be thought of as a higher order extension of linear interpolationp0p1p0p1p2p0p1p2p3Linear Quadratic CubicCurve Evaluationp0p1p2p3Find the point x on the curve as a function of parameter t:x(t)•de Casteljau Algorithmp0p1p2p3We start with our original set of pointsIn the case of a cubic Bezier curve, we start with four pointsde Casteljau Algorithmp0q0p1p2p3q2q1   322211100,,,,,,ppqppqppqtLerptLerptLerp   baba tttLerp  1,,de Casteljau Algorithmq0q2q1r1r0  211100,,,,qqrqqrtLerptLerpde Casteljau Algorithmr1xr0• 10,, rrx tLerpBezier Curvex•p0p1p2p3Recursive Linear Interpolation   322211100,,,,,,ppqppqppqtLerptLerptLerp  211100,,,,qqrqqrtLerptLerp 10,, rrx tLerp3210pppp210qqq10rrx3210ppppExpanding the Lerps                                               3221211010322121121101003232221211101001111111,,111,,111,,1,,1,,1,,pppppppprrxppppqqrppppqqrppppqppppqppppqtttttttttttttttLe rptttttttLer ptttttttLerptttLer ptttLerptttLerpBernstein Polynomials                     !!!133363133363363133333233223312330333223123023inininttintBttBtttBttttBttttBtBttttttttttttiinniiipxppppxThe Bernstein polynomial form of a Bezier curve iswhereandBernstein Polynomials   niinitBt0px     iinnittintB 1 !!!inininCubic Equation Form          010210321023010221033210333633313336333pdppcpppbppppadcbaxppppppppppxttttttCubic Matrix Form    0102103210233336333pdppcpppbppppadcbax ttt 32102300010033036313311ppppdcbadcbax tttCubic Matrix Form  zyxzyxzyxzyxpppppppppppptttttt333222111000233210230001003303631331100010033036313311xppppxMatrix Form CtxGBtxxBezBezzyxzyxzyxzyxppppppppppppttt3332221110002300010033036313311Bezier Curves & Cubic CurvesBy adjusting the 4 control points of a cubic Bezier curve, we can represent any cubic curveLikewise, any cubic curve can be represented uniquely by a cubic Bezier curveThere is a one-to-one mapping between the 4 Bezier control points (p0,p1,p2,p3) and the pure cubic coefficients (a,b,c,d)The Bezier basis matrix BBez (and it’s inverse) perform this mappingThere are other common forms of cubic curves that also retain this property (Hermite, Catmull-Rom, B-Spline)DerivativesFinding the derivative (tangent) of a curve is easy:cbaxdcbax  ttdtdttt 23223   dcbaxdcbax 01231223ttdtdtttTangentsThe derivative of a curve represents the tangent vector to the curve at some point tdtdx txConvex Hull PropertyIf we take all of the control points for a Bezier curve and construct a convex polygon around them, we have the convex hull of the curveAn important property of Bezier curves is that every point on the curve itself will be somewhere within the convex hull of the control pointsp0p1p2p3ContinuityA cubic curve defined for t ranging from 0 to 1 will form a single continuous curve and not have any gapsWe say that it has geometric continuity, or C0 continuityWe can easily see that the first derivative will be a continuous quadratic function and the second derivative will be a continuous linear functionThe third derivative (and all others) are continuous as well, but in a trivial way (constant), so we generally just say that a cubic curve has second derivative continuity or C2 continuityIn general, the higher the continuity value, the ‘smoother’ the curve will be, although it’s actually a little more complicated than that…Interpolation / ApproximationWe say that cubic Bezier curves interpolate the two endpoints (p0 & p3), but only approximate the interior points (p1 & p2)In geometric design applications, it is often desirable to be able to make a single curve that interpolates through several pointsPiecewise CurvesRather than use a very high degree curve to interpolate a large number of points, it is more common to break the curve up into several simple curvesFor example, a large complex curve could be broken into cubic curves, and would therefore be a piecewise cubic curveFor the entire curve to look smooth and continuous, it is necessary to maintain C1 continuity across segments, meaning that the position and tangents must match

View Full Document


School:
Email:
New Password:
Confirm Password:

UCSD CSE 167 - Curved Surfaces

Sign up for free to view:

Please select your school