Types of Curves/Surface

Explicit

An explicit representation of curve enables us to directly compute y at any value of x.

Surface equation in 3D space : x and y are independent variables and z is the dependant variable

y = f(x)

z = f( x, y)

y = mx + b  or x = c

z = - (Ax + By + D) / C

x = cos (3y +1/y)

z = sin(3x) * cos (xy) + ln(y)

Not Symmetric definition !

The explicit form is satisfactory when the function is single-valued and the curve has no vertical tangents

In the Cartesian plane, an explicit equation of a planar curve is given by y = f(x), where f(x) is a prescribed function of x. An explicit representation of curve enables us to directly compute y at any value of x. If we are asked to represent a straight line in the Cartesian coordinate by an explicit form, we will probably give the following equation:

y = kx + b,
provided that this line is not vertical to the x-axis. Otherwise, we have to represent vertical lines as x = c, where c is some constant value. Such problems inherent in explicit form is easily dealt with when solving a problem by hand. However, it is a nuisance when programming geometrical problems for a computer. Another drawback with respect to the use of explicit form is numerical stability. Referring to the above straight line, we note that the computation of y is numerically unstable if k goes to infinity, indicating the line is nearly vertical. In general, if a curve has nearly vertical tangents, we may expect overflow or rounding error problems when computing the function values. For these reasons, the use of explicit form in computer aided geometric design is very limited.

The explicit form is satisfactory when the function is single-valued and the curve has no vertical tangents. However, this precludes many curves of practical importance such as circles, ellipses and the other conic sections. An implicit equation of the form f(x,y) = 0 can avoid the difficulties of multiple values and vertical tangents inherent in the explicit form. For example, a unit circle with its centre at the origin is given by x2+y2-1 = 0. If we require an explicit equation of the same circle, then it must be divided into two segments, with y = +Ö[(r2-x2)] for the upper half and y = -Ö[(r2-x2)] for the lower half. This kind of segmentation creates cases which are a nuisance in computer programs.

Although an implicit form of curve can overcome some limitations inherent in an explicit form, it does not enable us to compute points on a curve directly. Usually, implicitly defined curves require the solution of a non-linear equation for each point and thus numerical procedures have to be employed. Furthermore, both explicit and implicit represented curves are axis-dependent. The choice of coordinate system directly affects the ease of use.

Explicitly and implicitly defined curves are sometimes called non-parametric curves. An alternative way of describing curves is the parametric form, which uses an auxiliary parameter to represent the position of a point. For example, a unit circle with centre at origin may be represented by an angle parameter u Î [0, 2p]:

x(u) = cosu,     y(u) = sinu.
Curves having parametric form are called parametric curves. A parametric curve that has a polynomial parameterization is called a polynomial curve, which is a standard in CAD systems for describing curves and surfaces. Since a point on a parametric curve is specified by a single value of parameter, the use of parametric techniques free us from dependence on any particular system of coordinates. Therefore, the parametric description of a curve enables coordinate transformations such as translation and rotation, required for graphical display, to be performed very simply. The parametric form also avoids problems which can arise in representing closed or multiple-valued curves and curves with vertical tangents in a fixed coordinate system. Furthermore, the parametric method lends itself to the piecewise description of curves and surfaces, which is a basic technique for the description of free form shapes. Due to these advantages, parametric curves are most commonly used in computer aided geometric design.