Arc Length and Curvature

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Section 14.3

• Arc Length and Curvature

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ARC LENGTH
Let a space curve C be given by the parametric
equations x f (t) y g(t) z
h(t) on the interval a t b, where f ', g',
and h' are continuous. If the curve is traversed
exactly once as t increases from a to b, then the
length of the curve is
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ALTERNATIVE FORMULA FOR ARC LENGTH
If
, is the vector equation of the curve
C, then the arc length formula can be written as
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THE ARC LENGTH FUNCTION
Suppose that C is a piecewise-smooth curve given
by a vector function r(t) f (t)i g(t)j
h(t)k, a  t b, and C is traversed exactly once
as t increased from a to b. We define its arc
length function s by
NOTE ds/dt r'(t)
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PARAMETERIZATION WITH RESPECT TO ARC LENGTH
Given the arc length function s(t) of a curve
given by r, it is often possible to solve for the
parameter t in terms of s. This allows us to
parameterize the curve with respect to arc length
by writing r as r r(t(s)). This is useful
because arc length arises naturally from the
shape of the curve and does not depend on a
particular coordinate system.
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CURVATURE
The curvature of a curve is where T is the unit
tangent vector,
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AN ALTERNATE CURVATURE FORMULA
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A THEOREM ABOUT CURVATURE
The curvature of the curve given by the vector
function r is
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CURVATURE OF A TWO-DIMENSIONAL PLANE CURVE
If y f (x) is a two-dimensional plane curve,
the curvature is given by
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THE PRINCIPAL UNIT NORMAL VECTOR
Given the smooth space curve r(t). If r' is also
smooth, we define the principal unit normal
vector N(t) (or simply unit normal) as where
T(t) is the unit tangent vector.
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THE BINORMAL VECTOR
The vector B(t) T(t) N(t) is called the
binormal vector. It is perpendicular to both T
and N and is also a unit vector.
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THE NORMAL AND OSCULATING PLANES

• The plane determined by the normal and binormal
  vectors N and B at the point P on a curve C is
  called the normal plane of C at P. It consists
  of all lines that are orthogonal to the tangent
  vector T.
• The plane determined by the vectors T and N is
  called the osculating plane of C at P. It is the
  plane that comes closest to containing the part
  of the curve near P. (For a plane curve, the
  osculating plane is simply the plane that
  contains the curve.)

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THE OSCULATING CIRCLE
The circle that lies in the osculating plane of C
at P, has the same tangent as C at P, lies on the
concave side of C (toward which N points), and
has radius ? 1/? is called the osculating
circle (or circle of curvature) of C at P. It is
the circle that best describes how C behaves near
P it shares the same tangent, normal, and
curvature at P.