9.4
The polar coordinate system is used to study non-rectangular regions in two dimensions, such as spirals and circles.
Imagine a plan position indicator on a ship displaying detected targets as dots relative to the ship at its center.
This location tracking uses the polar coordinate system.
The ship's position is taken as the pole, and a fixed direction serves as the polar axis.
A target location P is defined by its distance, r, from the pole and its angle, theta, from the polar axis.
To find the equivalent Cartesian coordinates for P(r, θ), imagine a right triangle where the radius, r, is the hypotenuse. Using trigonometry, the horizontal distance, x, is calculated as r cosine θ, and the vertical distance, y, as r sin θ.
To find polar coordinates from Cartesian values, the Pythagorean theorem gives r squared, and the tangent formula gives tangent theta.
When a target Q is located directly opposite P, its coordinates can be written as r, pi plus theta, or as negative r, theta. Here, the negative sign shows distance measured in the opposite direction, along the same line of sight.
The polar coordinate system provides a natural way to describe points in the plane when distances and directions are more meaningful than horizontal and vertical displacements. It is especially useful for modeling non-rectangular regions such as circles and spirals, where symmetry about a center point is easier to express than it is in a rectangular grid. A familiar example is a ship’s plan position indicator, which marks detected targets as dots positioned relative to the ship at the display’s center.
In polar form, the ship’s location is treated as the pole (the fixed reference point). A chosen fixed direction from the pole is the polar axis, analogous to the positive x-axis in Cartesian coordinates. Any target location P is specified by an ordered pair (r, θ), where r is the radius (the directed distance from the pole to the point), and θ is the angle measured from the polar axis to the line segment joining the pole to P. Angles are typically measured in radians in mathematical applications, though the geometric interpretation is the same in degrees.To convert (r, θ) to Cartesian coordinates (x, y), form a right triangle with hypotenuse r and acute angle θ at the pole. Trigonometric definitions then give x = r cosθ, y = r sinθ.
Polar coordinates are not unique. A point opposite P(r, θ) along the same line of sight can be expressed as (r, π + θ) or equivalently as (−r, θ), where a negative radius indicates measuring the distance in the opposite direction while keeping the same angular reference.
The polar coordinate system is used to study non-rectangular regions in two dimensions, such as spirals and circles.
Imagine a plan position indicator on a ship displaying detected targets as dots relative to the ship at its center.
This location tracking uses the polar coordinate system.
The ship's position is taken as the pole, and a fixed direction serves as the polar axis.
A target location P is defined by its distance, r, from the pole and its angle, theta, from the polar axis.
To find the equivalent Cartesian coordinates for P(r, θ), imagine a right triangle where the radius, r, is the hypotenuse. Using trigonometry, the horizontal distance, x, is calculated as r cosine θ, and the vertical distance, y, as r sin θ.
To find polar coordinates from Cartesian values, the Pythagorean theorem gives r squared, and the tangent formula gives tangent theta.
When a target Q is located directly opposite P, its coordinates can be written as r, pi plus theta, or as negative r, theta. Here, the negative sign shows distance measured in the opposite direction, along the same line of sight.
From Chapter 9:
Now Playing
Parametric Equations and Polar Coordinates
19 Views
Parametric Equations and Polar Coordinates
57 Views
Parametric Equations and Polar Coordinates
43 Views
Parametric Equations and Polar Coordinates
26 Views
Parametric Equations and Polar Coordinates
14 Views
Parametric Equations and Polar Coordinates
16 Views
Parametric Equations and Polar Coordinates
35 Views
Parametric Equations and Polar Coordinates
8 Views