which is a parametric form of the curve.
Perhaps the most known of the logarithmic spiral is that any line emanating from the origin the curve under a constant angle . This is seen e.g. by using the vector and its derivative , the latter of which gives the direction of the tangent line (see vector-valued function):
It follows that . The angle is called the polar tangential angle.
The logarithmic spiral (1) goes infinitely many times round the origin without to reach it; in the case one may state that
Letting one sees that the arc length from the origin to a point of the spiral is finite.
Any curve with constant polar tangential angle is a logarithmic spiral.
All logarithmic spirals with equal polar tangential angle are similar.
A logarithmic spiral rotated about the origin is a spiral homothetic to the original one.
The evolute of the logarithmic spiral is a congruent logarithmic spiral.
The catacaustic of the logarithmic spiral is a logarithmic spiral.
The families and are orthogonal curves to each other.
|Date of creation||2013-03-22 19:02:26|
|Last modified on||2013-03-22 19:02:26|
|Last modified by||pahio (2872)|