Where We Are

Unit III leaves flat 2D and dives into how 3D shapes are represented and shaded. Real surfaces are curved; computers prefer flat triangles. The art is in choosing the right approximation: polygon meshes for performance, parametric curves and surfaces (Hermite, Bezier, B-Spline) for design tools, and physically inspired illumination models for visual realism.

Why Parametric?

A curve described as y = f(x) has problems with vertical tangents and multi-valued y. A parametric curve P(u) = (x(u), y(u), z(u)) for u in [0,1] handles loops, vertical tangents, and 3D space uniformly. We choose cubic parametric pieces because cubics are the lowest-degree polynomials that allow inflection points and smooth joining (C1 / C2) between segments.

Why These Three Curve Families?

• Hermite: defined by endpoints and tangents - intuitive for engineers.
• Bezier: defined by control points whose convex hull bounds the curve - intuitive for artists.
• B-Spline: local control via knot vector - the workhorse of CAD and animation.

Why Two Shading Models?

• Gouraud: cheap, computes lighting at vertices and interpolates color across the polygon.
• Phong: higher quality, interpolates the normal across the polygon and computes lighting per pixel - captures specular highlights crisply.

Goals

By the end you can sketch a curve from control points, classify continuity, derive normals for shading, write the Phong illumination equation, and contrast Gouraud vs Phong shading visually.