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Other examples. 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.
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;
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
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.
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.
These examples are curves given by a Cartesian equation: f(x;y) = c where f: R2!R is a function in 2 variables: Put di erently, the curve is given as a level set C= f(x;y) 2R2 jf(x;y) = cg of the di erentiable function f. In our examples we have f(x;y) = y 2x, f(x;y) = x2+y2 and f(x;y) = y x2 respectively, and all curves are plane curves. If we ...
A rose curve is a graph that is produced from a polar equation in the form of: r = a sin nθ or r = a cos nθ, where a ≠ 0 and n is an integer > 1 They are called rose curves because the loops that are formed resemble petals.
