An Equiangular Spiral

An equiangular spiral, also known as a logarithmic spiral is a curve with the property that the angle α between the tangent and the radius at any point of the spiral is constant.
In polar coordinates:  r=ae^bθ   where a and b are positive real constants. 

In parametric form:  x(t)=ae^bt cos(t); y(t)=ae^bt sin(t).

The slider t allows us to change the parameter.

• Move the random point R over the spiral to see the constant angle α between the radius and the tangent.
• Use the sliders a and b to study the way they control the spiral.
• Consider a>0; a<0, b>0, b<0, b=0 and very large values of b.

Irina Boyadzhiev, Created with GeoGebra

Note: this construction is using the real constant b in the polar equation of the spiral as a slider parameter.

The following construction is very similar to the construction above. The difference is that the control b is replaced by the angle α

• Move the random point R over the spiral to see the constant angle α between the radius and the tangent.
• Use the sliders a and α to study the way they control the spiral.
• Consider a>0;  a<0;   α>90°,  α<90°, α=90°.

Irina Boyadzhiev, Created with GeoGebra