Thursday, January 21, 2016
Unizor - Geometry3D - Cylindrical Coordinates
Unizor - Creative Minds through Art of Mathematics - Math4Teens
Notes to a video lecture on http://www.unizor.com
We are familiar with polar coordinates on a plane. They are defined by a fixed point of origin (pole), a ray originated at this origin called polar axis and a polar angle or asimuth.
For any given point on a plane its polar coordinates consist of two numbers - non-negative distance from the pole called radius and non-negative (counterclockwise) angle from a polar axis to a ray connecting a pole with a given point.
Cylindrical coordinates are an expansion of polar coordinates on two-dimensional reference plane into a third dimension along the longitudinal or cylindrical axis perpendicular to a reference plane at the origin (pole). Traditionally, it is pictured as a horizontal reference plane with polar coordinates on it and a vertical cylindrical axis.
If we project any point A in three-dimensional space onto a reference plane, getting point Ap on this plane, the polar coordinates of this projection (radius ρ and asimuth φ) are the first two cylindrical coordinates of point A.
The length of a projection ApA with a sign corresponding to positive or negative direction from Ap to A relative to a direction of longitudinal axis is the third cylindrical coordinate called altitude or height, or z-coordinate (since it's similar to z-coordinate in the Cartesian system).
Thus, three coordinates, radius ρ, asimuth φ and altitude z, are cylindrical coordinates of a point in three-dimensional space.
They establish one-to-one correspondence between all points in space and three real numbers, ranging as follows: radius: ρ ≥ 0,
non-negative asimuth that is below 2π (radians), any altitude.
Let's use cylindrical coordinates to express certain properties of geometrical object.
1. A side surface of a cylinder of a radius R and height H with lower base lying on a horizontal reference plane with a center at the origin of coordinates can be defined by a system of one equation and two inequalities as follows:
ρ = R
z ≥ 0
z ≤ H
2. A plane going through a vertical (longitudinal) Z-axis and intersecting a horizontal reference plane at a line making an angle Φ with a polar axis on it can be defined by a very simple equation
φ = Φ
3. A side surface of a cone of a radius R and height H with lower base lying on a horizontal reference plane with a center at the origin of coordinates can be defined by a system of one equation and two inequalities as follows:
z = H·(1 − ρ/R)
ρ ≥ 0
ρ ≤ R