Curves and Curved Surfaces - lorentzo/IntroductionToComputerGraphics GitHub Wiki

Triangle is basic atomic rendering primitive for rasterization graphics rendering pipeline. Also, triangle is common rendering primitive for ray-tracing (sometimes we can define intersection with surface analytically so triangles do not have to be used).

Due to easier handling and authoring of some objects and animation paths, it is more convenient to use alternative geometric description.

Such alternative geometric description are curves and curved surfaces:

described precisely by equations
these equations are parametrized enabling easier handling for user (e.g., controlling only several points which affect the whole curve/surface rather than controlling all points of the curve/surface)
those equations can be evaluated and sets of triangles can be created for rendering purposes
more compact representation than triangles
provide scalable geometric primitives - surface can be turned into 2 or 2000 triangles enabling on the fly level of detail modeling depending on distance for surface
provide more smoother and continuous primitives than straight lines and triangles
animation and collision detection may become simpler
provide savings for model storage in terms of memory
transforming curved surfaces involves less matrix multiplications compared to mesh
less data that has to be sent from CPU to GPU (if this is possible by pipeline)
higher vertex density can be obtained if needed (smoother silhouettes and edges)
animation requires much smaller amount of points to be animated
collision detection can be simpler are more precise

Parametric curves

used to move viewer (camera), light or object along predefined path: changing both position and rotation (positional and rotational interpolation)
parametric curve describes points using some formula as function of parameter t -> p(t) - function delivers point for each t in [a,b]
- generated points are continuous (very small t can be used to get points which are very close to each other)

Bezier curves

Continuity and Piecewise Bezier curves

Cubic Hermite splines

Bezier curves are good for explaining theory behind the construction of smooth splines but sometimes are not predictable to work with
Hermit curve is defined by starting and ending points

B-Splines

uniform cubic B-splines

Parametric curved surfaces

natural extension of parametric curves
used to model objects with curved surfaces
this is parametric surface defined by small number of control points
by process of tessellation a parametric curve is transformed into triangle mesh for efficient rendering
adaptive number of triangles -> smooth shading and silhouettes
whole surface can be animated using small number of control points