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Saturday, 15 March 2014

Spiral




Spirals are of three kinds :-

1)  Equiangular Spiral or Logarithmic Spiral :-

Its equation is  r=ae to the power (theta cot alfa)

As   theta  ranges from  -infinity to  infinity.

r  ranges from   theta  to  infinity.

The curve has a characteristic property that the   tangent  makes a constant angle with the radius vector

i.e.  phay=alfa (constant)

origin is not on the curve.

2)  Spiral or Archimedes :-

The equation is  r=a theta

This is a curve described by a point which moves along a straight line with constant velocity while the line rotates with constant angular velocity about a fixed point in it.

Thus;  r=ut

and  theta= omega.t

Hence;  a=u/omega  

3)  Reciprocal Spiral.

r=a/theta.

Clearly,  y=r.sine theta

              =(r.sine theta.theta)/theta

            =(a.sine theta)/theta

therefore;  y=Lt theta-->0  (a.sine theta)/theta=a

therefore;   y=a   is any asymptote .