Drawing Parametric Curves Jean-Paul Mueller

Curves - The parametric form of a curve expresses the value of each spatial variable for points on the curve in terms of an independent variable, u: the parameter.

Curves - They can be developed for a wide variety of curves. - Are highly recommended when one wants to draw or analyze the curve

Curves - Parametric curves allow multiple values, so that curves can double back or even cross themselves.

Curves - A point on a curve is represented as a vector: p(u) = [x(u) y(u) z(u)] drawn as u varies.

Curves - One of the advantages of the parametric forms is that it is the same in 2 and 3 dimensions. - In 2 dimensions we just drop the equation for z. - The parametric form treats all 3 directions equally.

Curves - In 3 dimensions, we have 3 explicit functions. o x = x(u) o y = y(u) o z = z(u)

Curves - Parametric representations of curves are easy to join together. - This is good because usually you need more than one curve to create the shape you are trying to model.

Surfaces - Parametric surfaces require 2 parameters, which can be described as three equations of the form: o x = x(u, v) o y = y(u, v) o z = z(u, v)

Surfaces - Also, p(u, v) = [x(u, v) y(u, v) z(u, v)] represents a point on a surface.

Surfaces - As u and v take on values in specified range (0 to 1), the functions x, y, z trace out the location of the curve or surface. -The parametric form of curves and surfaces is the most flexible and robust for computer graphics.

Design Criteria - The way curves and surfaces are used in computer graphics and CAD is often different from the way they are used in other fields. - There are many considerations that determine why we use parametric polynomials of low degree, including: - Local control of shape - Smoothness and continuity - Ability to evaluate derivatives - Stability - Ease of rendering

Interpolation - How can we express a curve which has no simple mathematical definition? - We can draw an approximation to such a curve if we have an array of sample points.

Interpolation - Then we can guess what the curve should look like between the sample points.

Interpolation - If the curve is smooth and the samples are close enough together, we can make a pretty good guess as to what the missing portions should look like. - The guesses will probably not be exactly right, but it will be close enough for appearances.

Interpolation - Fill in the portions of the unknown curve with pieces of known curves which pass through the nearby sample points. -Then fit a portion of the unknown curve with a curve that we know.

Interpolation - Now fill in the gap between the sample points by finding the coordinates of points along the known curve.

Interpolation -These points then get connected with line segments.