matematicas visuales visual math

"For problems involving directions from a fixed origin (or 'pole') O, we often find it convenient to specify a point P by its polar coordinates

Polar coordinates | matematicasVisuales

where Polar coordinates | matematicasVisuales is the distance OP and Polar coordinates | matematicasVisuales is the angle that the direction OP makes with a given initial line OX, which may be identified with the x-axis of rectangular Cartesian coordinates. Of course, the point Polar coordinates | matematicasVisuales is the same as Polar coordinates | matematicasVisuales for any integrer n " (Coxeter, p.110)

We can use Polar coordinates for describing spirals, for example.

Coxeter introduce the Equiangular Spiral (or 'Logaritmic spiral') as the result of a continuous dilative rotation.

The polar equation of the Equiangular Spiral is

Equiangular Spiral: polar equation | matematicasVisuales

It receives the name of equiangular because the angle between the radius vector and the tangent is constant.

If phi | matematicasVisuales is the angle between the position vector OP and the tangent at P then (in general, in polar coordinates)

Equiangular Spiral: polar equation | matematicasVisuales

In the Equiangular spiral case, we can write

Equiangular Spiral: polar equation | matematicasVisuales

Calculating the derivative

Equiangular Spiral: derivative | matematicasVisuales

Then

Equiangular Spiral: derivative | matematicasVisuales

The angle between the position vector OP and the tangent at P is a constant and we can write the Equiangular spiral equation

Equiangular Spiral: derivative | matematicasVisuales

This spiral was called "spiral mirabilis" by Jacob Bernouilli.

Equiangular spiral: We can see that the angle formed by the radius vector and the tangent is constant | matematicasVisuales
We can see that the angle formed by the radius vector and the tangent is constant.
Equiangular spiral: We can see that the angle formed by the radius vector and the tangent is constant | matematicasVisuales

In the next applet you can play with a particular case: the Golden Equiangular Spiral. This spiral is related with the Golden Rectangle.

Equiangular spiral: the golden equiangular spiral | matematicasVisuales
This example is the golden equiangular spiral.

REFERENCES

Coxeter - Introduction to Geometry (John Whiley and sons)
Steinhaus - Mathematical Snapshots.
D'Arcy Thompson - On Growth and Form. (Cambridge University Press)

MORE LINKS

Equiangular spiral through two points
There are infinitely many equiangular spirals through two given points.
The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
The golden ratio
From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.
The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
The golden rectangle and the dilative rotation
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Durer and transformations
He studied transformations of images, for example, faces.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the dodecahedron.
Multiplying two complex numbers
We can see it as a dilatative rotation.