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UVA CS 445 - Bézier Curves

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CS 445 / 645 Introduction to Computer GraphicsSplines - HistoryBézier CurvesSlide 4Bézier vs. HermiteSlide 6Bézier Basis and Geometry MatricesBézier Blending FunctionsSlide 9Convex combination of control pointsSlide 11Why more spline slides?B-Spline Curve (cubic periodic)Uniform B-SplinesSlide 15B-spline Basis MatrixB-SplineSlide 18Nonuniform, Rational B-Splines (NURBS)Converting Between SplinesSlide 21Slide 22Rendering SplinesHorner’s MethodForward DifferenceSubdivision MethodsRendering Bezier SplineDo you want a 5th assignment?Slide 29Assignment 5Virtual TrackballTrackballSlide 33CS 445 / 645Introduction to Computer GraphicsLecture 23Lecture 23BBézier Curvesézier CurvesLecture 23Lecture 23BBézier Curvesézier CurvesSplines - HistoryDraftsman use ‘ducks’ and Draftsman use ‘ducks’ and strips of wood (splines) to strips of wood (splines) to draw curvesdraw curvesWood splines have second-Wood splines have second-order continuityorder continuityAnd pass through the And pass through the control pointscontrol pointsDraftsman use ‘ducks’ and Draftsman use ‘ducks’ and strips of wood (splines) to strips of wood (splines) to draw curvesdraw curvesWood splines have second-Wood splines have second-order continuityorder continuityAnd pass through the And pass through the control pointscontrol pointsA Duck (weight)Ducks trace out curveBézier CurvesSimilar to Hermite, but more intuitive definition of Similar to Hermite, but more intuitive definition of endpoint derivativesendpoint derivativesFour control points, two of which are knotsFour control points, two of which are knotsSimilar to Hermite, but more intuitive definition of Similar to Hermite, but more intuitive definition of endpoint derivativesendpoint derivativesFour control points, two of which are knotsFour control points, two of which are knotsBézier CurvesThe derivative values of the Bezier Curve at the The derivative values of the Bezier Curve at the knots are dependent on the adjacent pointsknots are dependent on the adjacent pointsThe scalar 3 was selected just for this curve The scalar 3 was selected just for this curve The derivative values of the Bezier Curve at the The derivative values of the Bezier Curve at the knots are dependent on the adjacent pointsknots are dependent on the adjacent pointsThe scalar 3 was selected just for this curve The scalar 3 was selected just for this curveBézier vs. HermiteWe can write our Bezier in terms of HermiteWe can write our Bezier in terms of Hermite•Note this is just matrix form of previous equationsNote this is just matrix form of previous equationsWe can write our Bezier in terms of HermiteWe can write our Bezier in terms of Hermite•Note this is just matrix form of previous equationsNote this is just matrix form of previous equationsBézier vs. HermiteNow substitute this in for previous HermiteNow substitute this in for previous HermiteNow substitute this in for previous HermiteNow substitute this in for previous HermiteMMBezierBezierMMBezierBezierBézier Basis and Geometry MatricesMatrix FormMatrix FormBut why is MBut why is MBezierBezier a good basis matrix? a good basis matrix?Matrix FormMatrix FormBut why is MBut why is MBezierBezier a good basis matrix? a good basis matrix?Bézier Blending FunctionsLook at the blending Look at the blending functionsfunctionsThis family of This family of polynomials is calledpolynomials is calledorder-3 Bernstein order-3 Bernstein PolynomialsPolynomials•C(3, k) tC(3, k) tkk (1-t) (1-t)3-k3-k; 0<= k <= 3; 0<= k <= 3•They are all positive in interval [0,1]They are all positive in interval [0,1]•Their sum is equal to 1Their sum is equal to 1Look at the blending Look at the blending functionsfunctionsThis family of This family of polynomials is calledpolynomials is calledorder-3 Bernstein order-3 Bernstein PolynomialsPolynomials•C(3, k) tC(3, k) tkk (1-t) (1-t)3-k3-k; 0<= k <= 3; 0<= k <= 3•They are all positive in interval [0,1]They are all positive in interval [0,1]•Their sum is equal to 1Their sum is equal to 1Bézier Blending FunctionsThus, every point on curve is Thus, every point on curve is linear combination of the linear combination of the control pointscontrol pointsThe weights of the The weights of the combination are all positivecombination are all positiveThe sum of the weights is 1The sum of the weights is 1Therefore, the curve is a Therefore, the curve is a convex combination of the convex combination of the control pointscontrol pointsThus, every point on curve is Thus, every point on curve is linear combination of the linear combination of the control pointscontrol pointsThe weights of the The weights of the combination are all positivecombination are all positiveThe sum of the weights is 1The sum of the weights is 1Therefore, the curve is a Therefore, the curve is a convex combination of the convex combination of the control pointscontrol pointsConvex combination of control pointsWill always remain within bounding region Will always remain within bounding region (convex hull)(convex hull) defined by control points defined by control pointsWill always remain within bounding region Will always remain within bounding region (convex hull)(convex hull) defined by control points defined by control pointsBézier CurvesBezierBezierBezierBezierWhy more spline slides?Bezier and Hermite splines have global influenceBezier and Hermite splines have global influence•One could create a Bezier curve that required 15 points to define the One could create a Bezier curve that required 15 points to define the curve…curve…–Moving any one control point would affect the entire curveMoving any one control point would affect the entire curve•Piecewise Bezier or Hermite don’t suffer from this, but they don’t Piecewise Bezier or Hermite don’t suffer from this, but they don’t enforce derivative continuity at join pointsenforce derivative continuity at join pointsB-splinesB-splines consist of curve segments whose polynomial consist of curve segments whose polynomial coefficients depend on just a few control pointscoefficients depend on just a few control points•Local controlLocal controlExamples of SplinesExamples of SplinesBezier and Hermite splines have global influenceBezier and Hermite splines have global influence•One could create a Bezier curve that required 15 points to define the One could create a Bezier curve that required 15 points to define the curve…curve…–Moving any one


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UVA CS 445 - Bézier Curves

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