TRANSLATION SURFACES

Translation surface can be created from the curve segment *k*, applying a class of translations determined
by a curve segment *h* (trajectory of the movement) (obr. 4.36).

**Synthetic representation:** (*k*, T_{P}(*v*))

**Analytic representation:**

**r**(*u*) = (*x*(*u*), *y*(*u*), *z*(*u*), 1),
*u* Î < 0, 1 >

**generating principle - ** class of translations determined by the curve segment *h*

**r***(*v*)=(*x**(*v*), *y**(*v*), *z**(*v*), 1),
*v* Î < 0, 1 >,

**modelled figure - ** translation surface patch

Obr. 4.37

Translation surface, created by a translation movement of the Lemniscate of Bernoulli on a sinusoidal segment is presented in the Obr. 4.37.

Special types of translation surface can be created from special curve segments *k*, *h*.

If *h* (trajectory of the movement) is a line segment, a patch of the cylindrical surface can be created,

or a patch of the prismatic surface.

Provided *k* and *h* are plane curve segments
located in one plane, a part of the plane can be determined.

Let the translation determined by the vector **a** = (*m*, n, *l*, 0) be given,
then **r***(*v*) = (*mv*, *nv*, *lv*, 1), *v* Î < 0, 1 >.

If *k * is a plane curve segment, *k *
a and vector **a** is the direction vector in the plane
a,

a surface patch can be created, which is a part of the plane
a (obr. 4.14).

Obr. 4.14 Obr. 4.15

If the curve segment *k* is a circle in the plane a and vector **a** is not the direction vector
in the plane a, a patch of the circular cylindrical surface can be created,
for the direction **a** ^ a a patch of the cylindrical surface of revolution.

Illustration of a cylidrical surface patch with the basic figure in the Archimedean spiral segment is in the Obr. 4.15.

If *k * is a polygon and vector **a** is not parallel to any of the polygon sides,
the modelled surface is a patch of the prismatic surface with edges in the direction of the vector **a**
(surface created from planar patches).