Thursday, January 21, 2016
Unizor - Geometry3D - Spherical Coordinates
Unizor - Creative Minds through Art of Mathematics - Math4Teens
Notes to a video lecture on http://www.unizor.com
Spherical coordinates is yet another way to identify a position of a point in three-dimensional space using three characteristics.
In Cartesian system of coordinates we use three linear dimensions to accomplish this (x, y, z), in cylindrical system we use two linear and one angular dimensions (ρ, φ, z), in spherical system of coordinates we use two angular and one linear dimension (φ, θ, r).
Here is how a spherical system of coordinates looks in a three-dimensional space relatively to Cartesian system:
The first characteristic is radial distance from the origin of coordinates (radius) r.
The Z-axis defines zenith direction and angle φ characterizes the deviation of the direction towards our point in space from zenith. It is called polar angle - the second (angular) characteristic in spherical system of coordinates.
The XY-plane is a plane of reference (similar to cylindrical coordinates) with X-axis serving as polar axis. The deviation from polar axis towards a projection of our point onto a reference plane is azimuth θ - the third (angular) characteristic in spherical system of coordinates.
Speaking about traditional letter designation, it must be noted that sometimes the letters φ and θ are used in opposite sense than is described above - letter θ designates a polar angle, while letter φ designates the asimuth.
Also, the polar distance sometimes is designated by a letter ρ.
Let's summarize the requirements of the spherical system of coordinates. We need a fixed point of origin O, an axis going through it that defines zenith direction (Z-axis on a picture above), from which we measure polar angle φ, a reference plane going through the origin perpendicular to zenith with an axis of azimuth=0 on it (XY-plane on a picture above with X-axis identifying azimuth of 0, Y-axis is not needed in this system of coordinates), from which we measure asimuth θ, and a unit of linear measurement to measure radial distance.
Given all this, to find the spherical coordinates of a point A in three-dimensional space, we have to make the following constructions and measurements:
(a) measure the radial distance r from the origin O to point A along ray OA;
(b) construct a plane through Z-axis and line OA and measure within this plane a polar angle φ of deviation from Z-axis, that is an angle from the positive direction of Z-axis to ray OA (it's the smaller angle that is used as a polar angle);
(c) project point A onto a reference plane into point Ap;
(d) within a reference plane measure an azimuth θ from the positive direction of the reference axis (X-axis on the picture above) counterclockwise to ray OAp.
The inversed procedure to find a point by its spherical coordinates is as follows:
(a) the radial distance r defines a sphere our point is located at;
(b) the polar angle φ defines a cone, on the surface of which our point is located; this cone intersects a sphere mentioned above at a circle;
(c) project the circle obtained above to a reference plane, also getting a circle of the same radius on this plane and, using azimuth θ, find a point on this circle - a projection of a point we have to find;
(d) from the projection point obtained above, going along the perpendicular to a reference plane at that point, we go to an intersection with a sphere and a cone constructed above to get the point we need.
Examples of usage of spherical coordinates.
1. A sphere of a radius R centered at the origin of coordinates is defined by a very simple equation with only radial distance r participates:
r = R
2. A conical surface with an apex at the origin of coordinates and an angle α between its generatrix and main axis of symmetry can be defined by a simple equation for polar angle φ: φ = α
3. Position of stars relatively to a position on the Earth can be expressed in spherical coordinates with zenith being on the axis of rotation of our planet, reference plane would go perpendicular to it on a parallel of an observer, almost constant polar angle would be a measure of deviation of the direction to a star from zenith and azimuth (deviation from some chosen direction, say, to the North on the reference plane) would change as our planet rotates around its axis.
A different specification of the direction to a particular star might be a time and an angle above the horizon. It's more practical, but at the core of it lies the same spherical system because the time is, actually, a measure of the Earth' rotation, that is, a replacement for azimuth, and an angle above horizon is a replacement for a polar angle.