Dogs at corners of a polygon chasing each other's tailEquiangular SpiralList of animations posted on this page.(Click the text to watch animation.)Equiangular Spiral  Zoom View Equilateral Triangle Square Regular Pentagon Regular Hexagon Ref.3 describes Equiangular Spiral as follows:
Suppose,after some elapsed time, the positions of the dogs that
started at A,B,C and D are A',B',C' and D' respectively.
It is evident from symmetry that A'B'C'D' will be a square and
the direction of the motion of each dog will be at constant angle, i.e. 45º to the
line joining it to the center of the courtyard.(a = 45º)
To create this drawing and animation:
Such a curve,in which the tangent at any point makes a constant angle with the radius drawn to that point from a fixed point is called an Equiangular ( or Logarithmic) Spiral. This curve was discovered in 1638 by
Rene Descartes (15961650)
during his study of dynamics.
Properties and formulas for equiangular spiralIn the drawing, cot a = MQ/MP,where MQ = dr , MP = r dq So cot a = dr/rdq or cot a dq = dr/r Integrating with repsect to q, log_{e}r = q cot a + log_{e}r_{0} where r_{0} is the value of r when q = 0. This may be written as log_{e}(r/r_{0}) = q cot a or r = r_{0} e ^{q cot a} where angle q is counted clockwise. ************ equiangular_spiral_def.dwg ************* In this example, the only known radius which can be used as a reference is the radius OA (= 1.0). So letting r_{0}= OA, and taking angle q starting from line OA counter clockwise, equation can be written simply as r = e^{q} , because cot a = 1 (1) All the radii are cut by the curve at a constant angle (a) (2) Arc length is the radius multiplied by constant, S = R/cos a
(3) Lengths of radii at equal angles to each other form a geometric progression. Further studies on the Equiangular Spiral curveThis drawing shows how the curve r = e^{q} looks like.The curve cuts x & y axis at P1,P2,P3,...and if the distance from the origin O is measured in the drawing, OP1 = 0.455488, OP2 = 0.094687, OP3 = 0.019683 Then OP1/OP2 = OP2/OP3 = 4.8105 = e^{p/2} It means that OP3P2 and OP2P1 are similar,i.e. when OP3P2 is rotated 90 degrees clockwise, and scaled up by the amount of e^{p/2}, then these two curve segments are identical.You can verify this by displaying this drawing and zooming up,copying a portion of the curve,and rotating 90. degrees, then scaling up by 4.8105. *********** equiang_spiral.dwg ************ This is the (3) property stated above. The other porperties can also be verified using this drawing. It is worth a trial. The whole spiral can be seen by running the Equiangular Spiral  Zoom View . To create this drawing and animation:
3 dogs case You can see the process in animation. To create this drawing and animation:
Angle a = p/6
4 dogs case You can see the process in animation. To create this drawing and animation:
Angle a = p/4
5 dogs case You can see the process in animation. To create this drawing and animation:
Angle a = 3p/10
6 dogs case You can see the process in animation. To create this drawing and animation:
Angle a = p/3

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Last Updated Aug15, 2006
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