Polar Coordinate System CALCULUS-III

1. Polar Coordinate SystemCALCULUS-III Dr. Farhana Shaheen

2. Polar Coordinate System • In mathematics, the polar coordinate system is a two-dimensionalcoordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. • The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.

3. 2-D (Plane) Polar Coordinates • Thus the 2-D polar coordinate system involves the distance from the origin and an azimuth angle. Figure 1 shows the 2-D polar coordinate system, where r is the distance from the origin to point P, and θ is the azimuth angle measured from the polar axis in the counterclockwise direction. Thus, the position of point P is described as (r, θ ). Here r & θ are the 2-D polar coordinates.

4. Figure: 1 • Any point P in the plane has its position in the polar coordinate system determined by (r, θ).

5. Some Points With Their Polar Coordinates

6. Rectangular and Polar Coordinates • Rectangular coordinates and polar coordinates are two different ways of using two numbers to locate a point on a plane. • Rectangular coordinates are in the form (x, y), where 'x' and 'y' are the horizontal and vertical distances from the origin.

7. A point in Cartesian Plane

9. Polar coordinates • Polar coordinates are in the form (r, θ), where 'r' is the distance from the origin to the point, and θ is the angle measured from the positive 'x' axis to the point:

10. Relation between Polar and Rectangular Coordinates • To convert between polar and rectangular coordinates, we make a right triangle to the point (x, y), like this:

11. The relationship between Polar and Cartesian coordinatesx = r Cos θ, y = r Sin θ

12. 1. Polar to Rectangular • From the diagram above, these formulas convert polar coordinates to rectangular coordinates: • x = r cos θ,   y = r sin θ. • So the polar point (r, θ) can be converted torectangular coordinates as: • (x, y) = ( r cos θ, r sin θ)  • Example: A point has polar coordinates: • (5, 30º). Convert to rectangular coordinates. • Solution:  (x, y) = (5cos30º, 5sin30º) • = (4.3301, 2.5)

13. Converting between polar and Cartesian coordinates • The two polar coordinates r and θ can be converted to the Cartesian coordinatesx and y by using the trigonometric functions sine and cosine: • while the two Cartesian coordinates x and y can be converted to polar coordinate r & θ , using the Pythagorean theorem) as follows:

14. 2. Rectangular to Polar • From the diagram below, these formulas convert rectangular coordinates to polar coordinates: • By the rule of Pythagoras: • r2 = x2 + y2. • Also, Tan θ = y/x  implies • θ = tan-1( y/x ) • So the rectangular point (x,y) can • be converted topolar coordinates • like this: • ( r,θ) = ( r, tan-1( y/x ) )  

15. To plot a point in Polar Coordinate • We first mark the angles, in the anti-clockwise direction from the polar axis.

16. Negative Distance

17. OQ is extension of OP • With coordinates P(r,θ) and Q(-r, θ+π)

18. For any real r > 0 and for all integers k:

19. A planimeter, which mechanically computes polar integrals • A planimeter is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.

20. Cartesian equations of Parabolas: • Move the original graph y=x2 up 2 units. The resultant graph is y= x2+2

21. Polar and Cartesian equations of a Parabola

22. Polar and Cartesian equations of a Parabola

23. Example: • Find the polar equation of each of the following curves with the given Cartesian equation: • a) x = c • B)x2 y + y3 = – 4  

24. Solution

25. To convert Cartesian equation into polar equation • Example:

26. Polar Equations of Straight Lines θ = α, for any fixed angle α. Exp: θ = π/4

27. Straight Lines • Standard equation • of straight line • in Cartesian coordinates: • y = mx + c

28. Polar Equations of Straight Lines • r Cos θ = k; or r = k Sec θ. • It is a vertical line through k. • It is equivalent to the Cartesian equation • x = k. • r Sin θ = k; or r = k Csc θ. • It is a horizontal line through k. • It is equivalent to the Cartesian equation • y = k.

29. Polar equation of a curve • The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of θ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as the graph of the polar function r. • Different forms of symmetry can be deduced from the equation of a polar function r. If r(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π − θ) = r(θ) it will be symmetric about the vertical (90°/270°) ray, and if r(θ − α°) = r(θ) it will be rotationally symmetric α° counterclockwise about the pole. • Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscates, limaçon, and cardioid.

30. Curve shapes given by polar equations • There are many curve shapes given by polar equations. Some of these are circles, limacons, cardioids and rose-shaped curves. • Limacon curves are in the form • r= a ± b sin(θ) and r= a ± b cos(θ) • where a and b are constants. • Cardioid (heart-shaped) curves are special curves in the limacon family where a = b. • Rose petalled curves have polar equations in the form of r= a sin(nθ) or r= a cos(nθ) for n>1. • When n is an odd number, the curve has n petals but when n is even the curve has 2n petals.

31. Polar Equations of Circles • r = k : A circle of radius k centered at the origin. • r = a sin θ : A circle of radius |a|, passing through the origin. If a > 0, the circle will be • symmetric about the positive y-axis; if a < 0, the circle will be symmetric about the • negative y-axis. • r = a cos θ: A circle of radius |a|, passing through the origin. If a > 0, the circle will be • symmetric about the positive x-axis; if a < 0, the circle will be symmetric about the negative • x-axis.

32. Equations of Circle • A circle with equation r(θ) = 1 • The general equation for a circle with a center at (r0, φ) and radius a is • This can be simplified in various ways, to conform to more specific cases, such as the equation • for a circle with a center at the pole and radius a.

33. A circle with equation r(θ) = 1

34. Parametric Equation of a Circle •     For a circle with origin (h,k) and radius r:      x(t) = r cos(t) + h       y(t) = r sin(t) + k

35. Graph Polar Equations • Step 1 • Consider r= 4 sin(θ) as an example to learn how to graph polar coordinates. • Step 2 • Evaluate the equation for values of (θ) between the interval of 0 and π. Let θ equal 0, π /6 , π /4, π /3, π /2, 2π /3, 3π /4, 5π /6 and π. Calculate values for r • by substituting these values into the equation. • Step 3 • Use a graphing calculator to determine the values for r. As an example, let • θ = π /6. Enter into the calculator 4 sin(π /6). The value for r is 2 and the point • (r, θ) is (2, π /6). Find r for all the (θ) values in Step 2. • Step 4 • Plot the resulting (r, θ ) points from Step 3 which are (0,0), (2, π /6), (2.8, π /4), (3.46,π /3), (4,π /2), (3.46, 2π /3), (2.8, 3π /4), (2, 5π /6), (0, π) on graph paper and connect these points. The graph is a circle with a radius of 2 and center at (0, 2). For better precision in graphing, use polar graph paper.

36. Simplify the Graphing of Polar Equations • Look for symmetry when graphing these functions. As an example use the polar equation r=4 sinθ. • You only need to find values for θ between π (Pi) because after π the values repeat since the sine function is symmetrical. • Step 2 • Choose the values of θ that makes r maximum, minimum or zero in the equation. In the example given above r= 4 sin (θ), when θ equals 0 the value for r is 0. So (r, θ) is (0,0). This is a point of intercept. • Step 3 • Find other intercept points in a similar manner.

37. Graphing Polar Equations • Example 1: Graph the polar equation given by r = 4 cos t • and identify the graph.

38. Solution • We first construct a table of values using the special angles and their multiples. It is useful to first find values of t that makes r maximum, minimum or equal to zero. r is maximum and equal to 4 for t = 0. r is minimum and equal to -4 for t = π and r is equal to zero for t = π/2.

39. Plotting of points in polar coordinates

40. Join the points drawing a smooth curve r = 4 cos t

41. Limacon • In geometry, a limaçon, also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.

42. Construction of a limacon

43. Polar Equations of Limacons • Equations of limacons have two general forms: • r = a ± b sin θand r = a ± b cos θ: • Depending on the values of a and b, the graph will take on one of three general shapes and will either pass through the origin or not as summarized below.

44. Equations of limacon • r = a ± b Cos θ; r = a ± b Sin θ • If |a| > |b| then you have a dimple; • If |a| = |b| then you have a cardioid; • If |a| < |b| then you have an interior lobe.

45. Graphs of Limacons • |a| >|b| |a| = |b| |a| < |b|

46. Cardioids • When |a| = |b|, the graph has a rounded \heart" shape, with the pointed (convex) indentation of the heart located at the origin. Such a graph is called a cardiod. They may be categorized as follows: • r = a(1 ±sin θ) . Symmetric about the positive y-axis if `+`; symmetric about the • negative y-axis if `- '. • r = a(1 ±cos θ) . Symmetric about the positive x-axis if `+'; symmetric about the negative x-axis if `-'. • In either case, the pointed \heart" indentation will point in the direction of the axis of symmetry. The maximum distance of the graph from the origin will be 2|a| and the point furthest away from the origin will lie on the axis of symmetry.

47. Limacons

48. Dimpled Limacons • r=3/2+cos(t) (purple)r'=3/2-sin(t) (red)

49. If |a| < |b| then you have an interior lobe in Limacon

50. The family of limaçons is varied by making arange from -2 to 2, and then back to -2 again.