PPT7: Conics and Circles CAP 6736 Geometric Modeling PPT7: Conics and Circles PPT and video are due no later than March 1 Submit to: lpiegl@gmail.com This template file is just an outline of the presentation that you need to complete. Additional pages may be necessary to fully explore the topic above. Each page should contain adequate text as well as illustrations. You are free to use all publicly available information (text as well as graphics) as long as the sources are properly acknowledged.

Team members’ contributions Member [name]:

Part I: Technical details For this part you will need an equation editor. You may use: MS equation editor, MathType, LaTeX, or Handwritten equations if all else fails

Various Forms of Conics Suggested content: Geometric definition of conics Classification of conics

Various Forms of Conics Suggested content: Various forms of conics ellipse hyperbola parabola

Rational Parametric Form of Conics Suggested content: Rational parametric form of conics Ellipse, hyperbola and parabola

Maximum Area Forms of Conics Suggested content: Maximum area forms of conics ellipse hyperbola parabola

Conics Forms in 3-D Suggested content: Conics in 3-D Ellipse, hyperbola and parabola

Rational Bezier Conics Suggested content: Rational Bezier form of conics Local coordinate frame Equation in the local frame

Rational Bezier Conics Suggested content: Type of conics Classification wrt. to shape invariance

The Shoulder Point Suggested content: Type of conics using the shoulder point

Circular Arc of less than 180° Suggested content: Circular arc as a Bezier conic

Infinite Control Points Suggested content: Use of infinite control points

Semi-circle via Infinite Control Points Suggested content: Semi-circle defined by infinite control points

Nine-point Circle Suggested content: The nine-point full circle Control points, knots and weights

Seven-point Circle Suggested content: The seven-point full circle Control points, knots and weights

Circular Arc of 240°using Negative Weight Suggested content: Circular arc of 240 degrees Control points, knots and weights

Circular Arc of 240° using Negative Weight and Knot Insertion Suggested content: Circular arc of 240 degrees Eliminating negative weights via knot insertion

Semi-circle Suggested content: Semi-circle Eliminating infinite control points via knot insertion

Seven-point Full Circle Suggested content: The seven-point full circle Control points, knots and weights

Circular Arcs of Various Sweep Angles Suggested content: Circular arcs of various sweep angles

Construction of Conics Suggested content: Conic constructions from end points and end tangents Parallel and non-parallel cases

Splitting Elliptical Arcs Suggested content: Splitting arcs for better convex hulls

Construction of Conics Suggested content: Conic construction examples

Form Conversion Suggested content: Form conversion General formulas

Form Conversion Suggested content: Form conversion: parabola Axis, focus and vertex

Form Conversion Suggested content: Form conversion: ellipse Center, major/minor axes and axis points

Form Conversion Suggested content: Form conversion: hyperbola Center, major/minor axes and axis points

Higher Order Circles Suggested content: Cubic circle Degree elevate semi-circle with infinite control points Control points, knots and weights

Higher Order Circles Suggested content: Cubic circle Degree elevate 240 arc circle with negative weight Control points, knots and weights

Higher Order Circles Suggested content: Degree four circle Full circle in complex plane Convert to Bezier with infinite control points Control points, knots and weights

Higher Order Circles Suggested content: Degree five circle Degree elevate degree four circle with infinite control points Control points, knots and weights

Biarcs Suggested content: Biarcs: definition Bezier formulation NURBS conversion: control points, knots, weights

Biarcs Suggested content: Biarcs: C-shaped and S-shaped

Biarcs Suggested content: Biarcs: parallel/anti-parallel end tangents

Biarc Approximation Suggested content: Biarc approximation Curvature plot

Part II: Design examples

Design Examples Suggested content: Add design examples: images and/or videos Give credit to the designers

Part III: GM lab For this part of the assignment you may use an existing system, such as Blender, or write the code and visualize the result using graphics tools like Processing.

Geometric Modeling Lab Suggested project: Design shapes using simple conics Font design using conics