Transformations
Between Surfaces
With Animations
Vesna Veličković
Eberhard Malkowsky
6th ISNM NSA NIŠ '2003
August 24 - 29, 2003
Faculty of Mechanical Engineering, Niš
Faculty of Science
and Mathematics,
University of Niš,
Srbija i Crna Gora
Surfaces of Revolution
Spherical and Pseudo-spherical
Surfaces of Revolution
Surface of revolution are
generating by rotation a curve
about the x3- axes.
x ( u )  ( r ( u ) cos u , r ( u ) sin u , h ( u ))
i
1
2
1
2
Surface of revolution with
constant Gaussian curvature
K>0 or K<0 are spherical or
pseudo-spherical surfaces of
revolution. (theory)
1
Spherical Surfaces of Revolution
r ( u )    cos
1
u
1
c
h (u ) 
1

1

2
c
2
 sin
2
u
1
du
1
c
where
  0,
c
1
K
,
K 0
Spherical Surfaces of Revolution
 c
 c
 c
Special case: sphere
 Hyperbolic spherical surface
of revolution
 Elliptic spherical surface of
revolution

(animation)
Pseudo-spherical Surfaces of
Revolution
K 0
K 
1
c
2
r ( u )  C 1 cosh
1
u
1
c
 C 2 sinh
u
1
c
Pseudo-spherical Surfaces of
Revolution
C1   C 2    0
C1    0
C2  0
C1  0
C2    0

Parabolic pseudo-spherical
surface of revolution

Hyperbolic pseudo-spherical
surface of revolution

Elliptic pseudo-spherical
surface of revolution
Classifications of Maps
A map F:SS* is called
 isometric if the length of every arc on S
is the same as that of its corresponding
image
 conformal or angle preserving if for
any pair of curves on S the angle between
them is the same as the angle between
their images.
 area preserving if any part of S is
mapped onto a part of S* with the same
surface area.
Isometric Maps
Theorem. A map F:SS* is isometric iff
their first fundamental coefficients gik and
g*ik with respect to the same parameters
(uj) and (u*j) satisfy
gik (uj) = g*ik (u*j) for i,k=1,2.
In particular, the Gaussian and geodesic
curvature of a surface are invariant under
isometric maps.
Since a sphere of radius r have Gaussian curvature
K = 1/r, and a plane K = 0, no part of a sphere
can be mapped isometrically into a plane.
Ruled Surfaces
Ruled surfaces play an important role in the
theory of isometric maps.
A ruled surface is a surface that contains a
family of straight line segments.
It is generated by moving vectors along
a curve.
Examples:
 Plane
 Cylinder
 Cone
 hyperboloid of one sheet
 hyperbolic paraboloid
Torse
A torse is a ruled surface which has the
same tangent plane at every point of
each of its generating straight lines.
Examples: planes, cylinders, cones.
Theorem. A surface of class Cr (r>1) is
part of a torse if and only if it has
identically vanishing Gaussian curvature.
Theorem. A sufficiently small part of a
surface of class Cr (r>2) can be mapped
isometrically into a plane if it is part of a
torse.
Conformal Maps
Theorem. A map F:SS* is conformal iff
their first fundamental coefficients gik and
g*ik with respect to the same parameters
(uj) and (u*j) are proportional, that is if
they satisfy
g*ik (u*j) = p(uj)gik (uj)
for i,k=1,2 and p>0.
Every isometric map is conformal.
Theorem. Every surface of class Cr (r>2)
can be mapped conformally into a plane.
Cartography
The problem of mapping a sphere to a
plane arises in cartography.
No length preserving maps exit from a
sphere to a plane.
Here we consider two conformal maps from
a sphere into a plane
 the stereographic projection (for pole
area)
 the Mercator projection (for equator
area).
Stereographic Projection
Stereographic Projection of
a Loxodrome on a Sphere
The stereographic projection of a loxodrome (a curve
on a surface that intersects each coordinate line in
one family at a constant angle) on a sphere is a
logarithmic spiral. This is clear since stereographic
projection is angle preserving.
The Mercator projection



Conformal
The images of the parallels and meridians of a sphere are
straight lines
The distance between the parallels increases as they
approach the poles.
Area Perserving Maps
Theorem. A map F:SS* is area preserving iff
the determinants g an g* of their first
fundamental coefficients gik and g*ik with
respect to the same parameters (uj) and
(u*j) are equal, that is if they satisfy
g* (u*j) = g (uj).
Theorem.
1. Every isometric map is area preserving
2. Every conformal and area preserving map is
isometric.
The Lambert Projection



Area preserving
The images of the parallels and meridians of a sphere are
straight lines
The meridians are equidistant and the distance between
the parallels decreases as they approach the poles.
Construction of
the Lambert Projection
From the Sphere to the Cylinder
Lambert Projection
From a Cylinder to a Plane
From a Circle to
a Straight Line Segment
Linear Transformation from
a Circle to a Straight Line Segment
Linear Transformation from
a Sphere to a Cylinder
Linear Transformation from
a Sphere to a Cone
Beyond Limits
From a Sphere to a Cylinder
and more...
The main purpose of our
software
We use our own software for geometry and
differential geometry [1].
The main purpose of our software is to visualize
the classical results in geometry and differential
geometry on PC screens, plotters, printers or
any other postscript device, but it also has
extensions to physics, chemistry,
crystallography and the engineering sciences.
To the best of our knowledge, no other
comparable, comprehensive software of this
kind is available.
[1] E. Malkowsky, W. Nickel, Computergrafik in der
Differentialgeometrie, Vieweg-Verlag Braunschweig
Wiesbaden,1993
The main concepts of our
software






Strict separation of geometry from the
technique of drawing.
Line graphics (contour).
Central projection.
Independent visibility check.
The software is open meaning that its
source files are accessible in Pascal. The
users may apply it in the solution of their
own problems. This makes it extendable
and flexible and applicable to research.
Object-oriented programming.