cardoid

cardoid - Polar coordinates Polar coordinates of a point P....

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Polar coordinates Polar coordinates of a point P. From polar to cartesian coordinates. From cartesian to polar coordinates. Polar equation of a curve. Examples Direction of a curve in polar coordinates Isogonal Curves Tangent line parallel to the polar-axis Tangent line orthogonal to the polar-axis Investigation of a curve ; an example From a cartesian equation t o a polar equation From a polar equation to a cartesian equation More examples of curves with polar equation Dot product in polar coordinates A line and its polar equation A line through the pole. A line d not containing the pole. A circle and its polar equation A circle with the pole as center. An arbitrary circle A special circle with A conic section and its polar equations Common points of two curves A problem with common points of two curves The cause of this problem A solution of this problem. A special property (P). Equations with property (P) The pole is a special point An extensive example. Rotating and the polar equation. Asymptotes in polar coordinates Visualization of dr/dt Asymptotes Area with polar coordinates General formula Area calculations Arc length in polar coordinates Length of a cardioid Polar coordinates of a point P. In the plane we choose a fixed point O, and we call it the pole. Additionally we choose an axis x through the pole and call it the polar axis.
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On that x-axis, there is just 1 vector E such that abs( E )=1. The pole and the polar axis constitute the basis of the polar coordinate system. Now, we take a point P. On the line OP we choose an axis u. The number t is a value of the angle from the x-axis to the u-axis. The number r is such that P = r. U The numbers r and t define unambiguous the point P. We say that (r,t) is a pair of polar coordinates of P. One point P has many pairs of polar coordinates. If (r,t) is a pair of polar coordinates, (r, t + 2.k.pi) is also a pair of polar coordinates and additionally (- r, t + (2.k+1).pi ) is a pair of polar coordinates too. Of course, k is an integer. Examples Figure 1: polar coordinates of P are (2 , 0.3) or (-2 , 3.44) or (2 , -5.98) . .. Figure 2: polar coordinates of P are (-1.4 , 3.6) or (1.4 , 0.46 ) or (1.4 , -5.8) . .. Figure 3: polar coordinates of P are (3 , 5.7) or (3 , -0.58) or (-3 , 2.56 ) . .. The polar coordinates of the pole O are by definition (0,t) with t perfectly arbitrary. From polar to cartesian coordinates. A polar coordinate system is given and point P has polar coordinates (r,t). We choose a y-axis through the pole O and perpendicular to the x-axis. So, we have cartesian axes x and y. Call (x,y) the cartesian coordinates of P. According to the previous definition, the cartesian coordinates of U are (cos t, sin t).
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cardoid - Polar coordinates Polar coordinates of a point P....

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