Curvature of a discrete curve in 3D space
Are Mjaavatten, March 2020
are@mjaavatten.com
Assume that the curve is given as list of x, y and z coordinates and let
. The list is
ordered so that moving along the list corresponds to moving along the curve. The circle passing
through all three corners of the triangle formed by the neighboring points
and
is called
the circumscribed circle of the triangle. Let
be the center and
the radius of this circle. The
curvature of the curve at
is defined as:
( 1 )
We define the curvature vector k
i
as the vector of length
in the direction from
to
. In the
next paragraph we develop equations for the circumscribed circle for a general triangle ABC.
Center of circle through three given points in 3D space
A
C
B
M
c/2
b/2
α
α
α
e
1
e
2
Figure 1
Consider a triangle in 3D space with corners A, B and C. The three points define a plane and Figure 1
shows the triangle in this plane. The center M of the circumscribed circle is equally distant from all
corners and is therefore located at the intersection of the normals at the midpoint of sides AC and
AB. The length of the side opposite to point B is denoted b- Likewise, c the length of side AB. Let α
be the angle between sides AB and AC. Let the boldface symbols A, B and C denote the vectors to
the corner points in some Cartesian coordinate system.
The cross product
( 2 )
is normal to the triangle plane, pointing towards the viewer. The norm of D is twice the area of the
triangle:
( 3 )
