Lecture
Let us now move from cubic curves to bicubic surfaces defined
by cubic equations in two variables s and t. By varying both parameters from 0 to 1, one can determine all points on a patch of the surface. If one of the parameters
is assigned a constant value while the other is varied over the range 0 – 1, the result is a cubic curve. Just as in the case of curves, we will consider the equation only for x:
or in algebraic form:
where
denotes the transpose of matrix 𝑇.
𝐶𝑥 specifies the coefficients of the bicubic polynomial. There are likewise 𝐶𝑦 and 𝐶𝑧 specifying
the coefficients of 𝑦(𝑠, 𝑡) and 𝑧(𝑠, 𝑡).
To express the surface in Hermite form, an approach analogous to
the one presented for bicubic curves is applied, which allows control points and tangent vectors to be used to determine the coefficients of the bicubic polynomial.
Let us consider the equation of the bicubic curve from section 2.1(13), in which t is replaced by s.
Let us rewrite this equation so that the Hermite geometry matrix is not a constant but a function of t.
The functions P1x(t) and P4x(t) describe the x-components of the start and end points
of the curve defined by the parameter s. For each value of t, two
endpoints are determined. Similarly, R1x(t) and R4x(t) describe the tangent vectors at the end
points of the cubic curve constructed as a function of 𝑠.
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