In three dimensions, the points of constant x are planes they intersect the planes of constant y or constant z at right angles.ĭespite these obvious advantages, many problems in physics are handled advan-tageously in other coordinate systems that may reflect some symmetry of the physical system under study. Harris, in Mathematics for Physical Science and Engineering, 2014 6.3 Curvilinear Coordinate SystemsĬartesian coordinate systems have two great virtues: (1) they are uniform, with the local geometry identical at all points, and (2) they are orthogonal in two dimensions, the lines of constant x are perpendicular to the lines of constant y. Also, thelarger the value of a, thesmoother the curve becomes.Frank E. When a is larger than 1, the “ties” disappear. In the polarequation r = a + b cos (k ), there areexactly k “petals”.Besides, when a = 1, the petals “tie” at the origin. It changes from a limacon to a “flower-petal” shape. Recallthe shape of r = a + b cos ( ) is a limacon.From the two graphs above, we can see that if is multiplied by k then the shape of y = a + b cos ( ) changescompletely. First, let a= 1, 2, and 5 with b = 1 (again, b only affects the size of the curves), andexplore the curves when k = 2, 3, 4, and 5. Now,it’s time to explore r = a + b cos (k ). Please look at the graphs below for the differences. When k is an even number, there are 2k petals in a rose. When k is an odd number, there are k petals in a rose curve. Thereis an interesting finding here, the number of petals of a rose is determined bywhether k is an odd numberor an even number. Therefore, wewill use only positive k inthe rest of our exploration.

It seems likewhether k is positive ornegative, it has nothing to do with the shape of the rose curves. Since b only determines the location and the size of r = bcos (k ), let b = 1 to simplify our exploration.įromthe two graphs above, we can see that multiply by k form petal shape curves. Howabout multiply by a number kto alter the graphs of the two polar equations above? Let’s exam the polarequation r = b cos ( k ) first. When a is larger than 2, the curves just curve outward without inner loops or dimples.All the curves form by the polar equation r = a + cos ( ) are called limacon (pronounced lee muh SOHN).

When 1 a 2, it forms curves that have dimpleslike the red, the dark blue, and the light blue ones. When a is smaller than 1, the polar equation r = a + b cos ( ) forms curvesthat have inner loops like the green and the pink ones below. R= a + b cos ( ) below are curves with different shapes. Next, we are going to explore the polargraph of r = b cos ( ) by adding thefunction by a number, say a. Nowwe know that we can alter the position and the size of a polar graph bymultiplying the function by a number. When b is positive, the circles are on the right of they-axis and the circles are on the left of the y-axis when b is negative. Also, the positive and negative signs of b determine the locations of the circles. Notice that this graph arecircles with diameters of b unit and pass through the origin. Let’sstart with the simplest polar equation r = bcos ( ). I amgoing to explore some polar equations and their polar graphs in the followingsections. A plot of all points whose coordinates (r, ) satisfy agiven polar equation is called a polar graph.

For example, r =3 cos ( ) is a polarequation. As mentioned earlier, distances and angles are twomain elements of the polar coordinate system and therefore a polar equation canbe specified by defining r (distances) as a function of (angles) in many cases. Note that theangle,, can bemeasured in degrees or radians.Īnequation expressed in terms of polar coordinates is called a polar equation. If a point P has polarcoordinates (r, ), then thedistance from the pole to point P is | r | and the angle formed by OP and thepolar axis is. A horizontal ray directed from the poletoward the right is called the polar axis. Inthe polar coordinate system above, a fixed point O is called the pole (origin). Since distances and angles are the main componentsof the polar coordinate system, this system is especially useful under thecircumstances where the relationship between two points is most easilyexpressed in terms of angles and distances. The polarcoordinate system is atwo-dimensional coordinate system in which the position of an object isrecorded using the distance from afixed point and an angle made with afixed ray from that point.

Polar coords