Applied Graphics and Geometry
Interactive Examples (AGGIE-Volume 1)
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Department of Mechanical Engineering

Each lecture below contains a Task, Theory, Solutions and Review Questions that
complement classroom lectures and self paced study. Mathcad 2001i Professional-Academic edition is used.

Users external to USNA will need Mathcad 2001i or higher to read the .mcd files.

Scroll screen down to view the entire Index and click on lecture of choice.

Preface(preface.htm)

Chapter 1:  Graphic Transformations

•  Rotation of Data to Create Orthographic Views at Specified Eye Position(I-1.htm)
•  Rotation of Data to Create Orthographic Views from Normal Eye Position(I-2.htm)
•  Isometric Projection(I-3.htm)
• Use of Homogeneous Coordinates to Create Orthographic Views(I-4.htm)
• Translation and Rotatiion using Homogeneous Coordinates(I-5.htm)
• Rotations and Projections(I-6.htm)
• Noncommutative Nature of 3D Rotations(I-7.htm)
• Rotation and Translation(I-8.htm)
• Projections of a Trapezoid(I-9.htm)
• Interpretatiion of Surface Plots(I-10.htm)
• Trimetric Projection(I-11.htm)
• Isometric and Dimetric Projections(I-12.htm)
• Simple Origami of Triangular Facets(I-13.htm)
• Rotation about a Local Axis(I-14.htm)
• Reflection through a Plane(I-15.htm)
• Right Hand and Left Hand Coordinates(I-16.htm)
• Local Scaling and Transformation(I-17.htm)
• Overall Scaling(I-18.htm)
• Local Shearing(I-19.htm)
• Order of 3-D Rotations(I-20.htm)
• Rotation about Parallel Principal Axes(I-21.htm)
• Multiple Rotations about Parallel Principal Axes(I-22.htm)
• Perspective Transformations(I-23.htm)
• One and Two Point Perspectives of I-Beam(I-24.htm)
• Perspective Views and Vanishing Points of a Prism(I-25.htm)
• Stereo Pair(I-26)
• Arbitrary Plane of Projection(I-27.htm)
• Three Dimensional Reconstruction(I-28)
• Transformation Reconstruction(I-29)
• Transformation Reconstruction using Two Photographs(I-30)
• Chapter 2:  Geometric Modeling
• Normal to a Plane(II-1.htm)
• Edge View of a Plane(II-2.htm)
• Planes in True Size(II-3.htm)
• Assembly of Line and Planes(II-4.htm)
• Line seen as a Point(II-5.htm)
• Line seen in True Length(II-6)
• Parallel Lines(II-7.htm)
• Implicit Plane parallel to a Plane(II-8.htm)
• Plane containing Points and parallel to a Line(II-9.htm)
• Shortest Distance form Point to Line(II-10.htm)
• Shortest Distance from Point to Plane(II-11.htm)
• Plane perpendicular to Line(II-12.htm)
• Shortest Distance from Origin to Plane(II-13.htm)
• Intersection between Line and Plane(II-14.htm)
• Parametric Equations for a Plane and Line(II-15.htm)
• Projection of Line onto a Plane(II-16.htm)
• Shortest Distance between Skew Lines(II-17.htm)
• Shortest Distance between Two Lines with Given Slope(II-18.htm)
• Vertical Distance between Two Skew Lines(II-19.htm)

Chapter 3: Geometric Construction

• Circle in 3D Space with given Diameter(III-1.htm)
• Circle in 3D Space through Three given Points(III-2.htm)
• Circular Fillet in 3D Space(III-3.htm)
• Ellipse in 3D Space(III-4.htm)
• Ellipse defined by Internal Rectangle(III-5.htm)
• Ellipse in 3D Space using Conjugate Diameters(III-6.htm)
• Tangents drawn to Ellipse(III-7.htm)
• Parabola in 3D Space through Three Points(III-8.htm)
• Bezier Parabola(III-9.htm)
• Bezier Parabola through Three Specified Points(III-10.htm)
• Hyperbola in 2D Space(III-11.htm)
• Shortest Distance from Point to Helix(III-12.htm)
• Shortest Distance from Point to 3rd Order Bezier Curve(III-13.htm)
• Bezier Surface of Revolution(III-14.htm)
• Intersection of 3D Lines(III-15.htm)
• Apparent Angle of Intersection between Two Lines(III-16.htm)
• Intersection of Line with Parametric Surface(III-17.htm)
• Shortest Distance from Point to 4th Order Bezier Curve(III-18.htm)
• Intersection between Bezier Curve and Hyperbola(III-19.htm)
• Cardioid in Polar Coordinates(III-20.htm)
• Intersection between Cardioid and Circle(III-21.htm)
• Intersection between Two Circles(III-22.htm)

Chapter 4: Curves and Surfaces

• Parabola and Hyperbola Intersection(IV-1.htm)
• Ruled Surface between Parabola and Hyperbola(IV-2.htm)
• Area, Centroid and Moment of Inertia Computations(IV-3.htm)
• Orthographic Projections of Surface(IV-4.htm)
• Ruled Surface between Circle and Ellipse(IV-5.htm)
• Surfaces of Revolution(IV-7.htm)
• Helical Surface(IV-8.htm)
• Surface Patch Derivatives on Unit Sphere(IV-9.htm)
• Gaussian and Average Curvature(IV-10.htm)
• Surface Area and Volume(IV-11.htm)
• Parametric Surface Mapping(IV-12.htm)
• Intersection of Line with Sphere(IV-13.htm)
• Geometric Properties of Paraboloid of Revolution(IV-14.htm)
• Intersection between Plane and Surface(IV-15.htm)
• Intersection between Plane and Cone(IV-16.htm)
• Sweep Surface with Axis Rotation(IV-17.htm)
• Sweep Surface with Circular Cross Section(IV-18.htm)
• Bezier Surface of Revolution about Vertical(IV-19.htm)
• Bezier Surface of Revolution about Horizontal(IV-20.htm)
• B-Spline Surface of Revolution(IV-21.htm)
• Piecewise Cubic Spline Interpolation(V-1.htm)
• Normalized Cubic Spline(V-2.htm)
• Relaxed End Condition(V-3.htm)
• Normalized Cubic Spline and Geometric Properties(V-4.htm)
• Chord Length Parameters(V-5.htm)
• Conic Bezier Curve(V-6.htm)
• Bezier Blending Functions(V-7.htm)
• Sixth Order Bezier Curve(V-8.htm)
• 3rd Order B-Spline Curve(V-9.htm)
• 2nd Order Open Uniform B-Spline(V-10.htm)
• 4th Order Open Uniform B-Spline(V-11.htm)
• 4th Order Periodic B-Spline(V-12.htm)
• Closed 4th Order Periodic B-Spline(V-13.htm)
• Periodic B-Spline of Airfoil Shape(V-14.htm)
• Closed 4th Order Non-Symmertical Periodic B-Spline(V-15)
• Matrix Formulation for Summetric B-Spline(V-16)
• Derivatives of 2nd Order Open Uniform B-Spline(V-17.htm)
• Derivatives of 4th Order Open Uniform B-Spline(V-18.htm)
• Five Vertex 2-D B-Spline Curve Fit(V-19.htm)
• B-Spline Curve Fit through 11 Points(V-20.htm)
• B-Spline Curve Fit through 9 Points(V-21.htm)
• B-Spline Curve Fairing(V-22.htm)
• Four Vertex Curve Fairing with Third Order Open Uniform B-Spline(V-23.htm)
• Regular and Rational 3rd Order B-Spline Curves(V-24.htm)
• 4th Order Regular and Rational B-Spline Curve(V-25.htm)
• 3rd Order Regular and Rational Periodic B-Spline Curves(V-26.htm)
• 4th Order Regular and Rational Periodic B-Spline Curves(V-27.htm)
• 2nd Order Open Rational Uniform B-Spline Derivatives(V-28.htm)
• 4th Order Open Uniform Derivatives(V-29.htm)
• 3rd Order Regular and Rational Open Uniform B-Spline Curves(V-30.htm)
• 3rd Order Non-Uniform Knot Vector for Rational Conics(V-31.htm)
• 3rd Order Non-Uniform Knot Vector for Closed Circle(V-32.htm)