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After studying this chapter you should. be familiar with cartesian and parametric equations of a curve; be able to sketch simple curves; be able to recognise the rectangular hyperbola; be able to use the general equation of a circle; be able to differentiate simple functions when expressed parametrically.
r = cos( ) + sin( ) + sec( ) + csc( ) + cot( ) + tan( ) This curve has Cartesian equation ((x2 + y2)(xy x y) xy(x + y))2 (x2 + y2)3 = 0. and asymptotes x = 2 and y = 2. ln. This is a spiral. Exercises: Explain why the spiral gets more and more tightly wound as it goes farther from the origin.
The (x, y) co-ordinates of a point in the plane are called its Cartesian co-ordinates. But there is another way to specify the position of a point, and that is to use polar co-ordinates (r, θ). In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again.
A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points in the xy-plane, using the origin (0; 0) and the positive x-axis for reference.
2. Simple curves in polar coordinates We are used to describing the equations of curves in Cartesian variables x,y. Thus x2 + y2 = 1 represents a circle, centre the origin, and of radius 1, and y = 2x2 is the equation of a parabola whose axis is the y-axis and with vertex located at the origin. (In colloquial terms the vertex is the ‘sharp
Graphing a polar equation is accomplished in pretty much the same manner as rectangular equations are graphed. They can be graphed by point-plotting, using the trigonometric functions period, and using the equation’s symmetry (if any).
point. Every point can be represented by its Cartesian coordinate. 1.1.3. What is the meaning of a curve in R2? Intuitionally, this is a subset of R2 with points in it can move in 1 free degree. Examples from pictures: • smooth curves vs curves with sharp points; • simple curves vs multiple curves.
