Todays Program

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Todays Program

• Curves Determination
• Analytic Description of Curve
• Plane Curves
• Space Curves
• Parametric Curves
• Transformation of Parameter
• Specific Points On Curve
• Length of Curve
• Osculating, Normal and Rectifying Planes
• Frenet Trihedron
• Curvature, Osculating Circle
• Examples

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Curves
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Curves Determination
Curves as models of real objects that have one
dominant dimension and two remaining dimensions
are negligible cable, wire, rails...

• Empirical Curves
• Graph of Function
• Intersection of Two Surfaces
• Curves Defined by Equation
• Curves for CAD

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Analytic Description of Curve
Planar Curve

• Parametric Form X(t) x(t)y(t)0
  x(t)y(t), t?I
• Explicit Form (graph of a function) yf(x) ? X(t)
  ty(t)0, t?I
• Implicit Form f(x,y)0
• Polar Coordinates ?f(f) ? X(f) ?.cos f ?.sin
  f0, f?I

parabola yx2 ? X(t) t t20, t?R
circle x2 y2 -10
Archimedes Spiral rk.f, k?0, k?R
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Analytic Description of Curve
Spatial Curve

• Parametric Form X(t) x(t)y(t)z(t), t?I
• Intersection of Two Surfaces

Helix X(t) r.cos tr.sin tvot, t?R
Line as a section of a plane x-z-10 and a plane
yz-10
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Parametric Curve

• Vector Valued Function X I?R ? R3
• X(t) x(t), y(t), z(t), t ? I, x, y, z real
  valued function
• x, y, z continuous in I
• X(t) (x(t), y(t), z(t)) ? (0, 0, 0) for all
  t ? I
• X is called parameterized curve.

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Transformation of Parameter
Parabola K X(t) t t2, t?I Transformation t
v 2, v?J Curve Q Y(v) X(v2) v 2
v24v4, v?J
K X(t), t?I Q Y(v), v?J
K Q
Two parameterizations of one geometric image of
the curve.
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Specific Points on Curve
Parameterized curve X(t) x(t), y(t), z(t), t
? I. Singular point X(t0)(0, 0, 0) or X(t0)
does not exist. of the first order there
exists a reparameterization Y(v) such that
X(t0)Y(v0) and Y(v0) exists and Y(v0)?(0, 0,
0), of the second order in other cases.
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Specific Points on Curve
Parameterized curve X(t) x(t), y(t), z(t), t
? I. Inflection point X(t), X(t)
exist X(t0), X(t0) are linearly dependent.
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Length of Curve
Length s of a curve K set by X(t) between points
aX(ta) and bX(tb)
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Osculating, Normal and Rectifying Plane
Curve X(t) x(t), y(t), z(t). Tangent vector
X(t) (x(t), y(t), z(t)) X(t) (x(t),
y(t), z(t)) X(t) non-inflection point
(X(t), X(t) are linearly independent). Normal
plane ? ? tangent t. Osculating plane ? X(t0),
X(t0), X(t0). Principle normal n ? ? ?
?. Rectifying plane ? ? principle normal
n. Binormal b ? ? ? ?.
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Frenet Trihedron
Frenet trihedron is an ordered triple of vectors
T, N, B.
If a planar curve is in the explicit form yf(x),
then
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Curvature
If a curve K is set by parameterization X(t),
where t is a general parameter, then the
curvature of the curve K in the point X(t) is
If a planar curve is in the explicit form yf(x),
then
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Osculating Circle
A circle that lies in the osculating plane of the
point TX(to) on the curve K and that has the
centre S lying on the principal normal n of the
point T and r r(to)1/k(to) far from T, is
called osculating circle of the curve K in the
point T. The osculating circle and the curve
have the same tangent and the same curvature.
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Osculating Circle

• Osculating Circle in TX(to)
• Radius r r 1/k(to)
• Centre S SX(to)r N(to), where N(to) is unit
  vector of the principle normal n in T.

Implicit equation of osculating circle for planar
curve (x-s1)2 (y-s2)2 r2.
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Osculating Circle
Ex. Determine an equation of the osculating
circle for the cubic curve yx3/3 in the point
T1,?.
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Osculating Circle
Ex. Determine an equation of the osculating
circle for the cubic parabola yx3/3 in the point
T1,?.
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Example
Ex. Determine the function of the curvature of
the parabola yx2.
Ex. Determine the function of the curvature of a
helix.
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Osculating Circles of Ellipse
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Osculating Circles of Cycloid