-T, N. de (i) The concept of curvature and radius of curvature are defined by extending those con- cepts from circles to curves in general. The curvature & is defined as the rate (with respect to length along the curve) of rotation of the tangent vector. Show that the definition of curvature given in class, namely lim 1-²₂ T(t₁) - T(t₂)| |l(t₁) - l(t₂)| dT de is in fact the rate of rotation of T (i.e., that it gives the rate of change of the angle T makes with a fixed direction). Show also that for a circle in the plane k = 1/R where R is the radius of the circle.
-T, N. de (i) The concept of curvature and radius of curvature are defined by extending those con- cepts from circles to curves in general. The curvature & is defined as the rate (with respect to length along the curve) of rotation of the tangent vector. Show that the definition of curvature given in class, namely lim 1-²₂ T(t₁) - T(t₂)| |l(t₁) - l(t₂)| dT de is in fact the rate of rotation of T (i.e., that it gives the rate of change of the angle T makes with a fixed direction). Show also that for a circle in the plane k = 1/R where R is the radius of the circle.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We have shown in class that the acceleration of a particle can be decomposed into components
that are tangential and normal to its trajector as
a = kv²N+ÏT
where T and N are the unit tangent vector and principal (unit) normal vector, respectively,
is the length along the trajectory, v = is the speed, and is the curvature of the trajectory.
The curvature is related to the radius of curvature p by k = 1/p.
The plane formed by T and N is called the osculating plane, and can be thought of as the
instantaneous plane of the trajectory. The unit vector
B = TX N
is called the binormal vector, and by definition is always perpendicular to the osculating
plane. We have shown in class that both T and B are parallel to N.
The torsion r(t) of the trajectory is defined by
or, equivalently, by
B = -TIN
dB
de
= -T, N.
(i) The concept of curvature and radius of curvature are defined by extending those con-
cepts from circles to curves in general. The curvature & is defined as the rate (with
respect to length along the curve) of rotation of the tangent vector. Show that the
definition of curvature given in class, namely
|T(t₁) - T(t₂)|
lim
t₁-t₂|l(t₁) -l(t₂)|
d'T
de
= K,
is in fact the rate of rotation of T (i.e., that it gives the rate of change of the angle T
makes with a fixed direction). Show also that for a circle in the plane = 1/R where
R is the radius of the circle.
(ii) Prove the following Frenet-Serret formulae:
T = KUN,
N=-KUT + TUB.
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