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vb.net generate qr code Spherical Coordinates in .NET framework
Spherical Coordinates Code39 Scanner In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Code39 Printer In .NET Framework Using Barcode maker for .NET framework Control to generate, create Code 3 of 9 image in .NET framework applications. We can have nonstandard vertical angles, although things are simplest if we keep them nonnegative but no larger than p Theoretically, all possible locations in space can be covered if we restrict the vertical angle to the range 0 f p If it s outside this range, such as p < f < 0 or p < f < 2p, we can multiply the radius r by 1, and then add or subtract p to or from f, and we ll end up at the point we want But those are confusing ways to get there! The radius r can be any real number, but things are simplest if we keep it nonnegative If our horizontal and vertical direction angles put us on a ray that goes from the origin through P, then r > 0 If our direction angles put us on a ray that goes from the origin away from P, then r < 0 We have r = 0 if and only if P is at the origin If we find ourselves working with a negative radius, we should reverse the direction by adding or subtracting p to or from both angles, keeping 0 q < 2p and 0 f p Then we can take the absolute value of the negative radius and use it as the radius coordinate Code39 Recognizer In .NET Framework Using Barcode reader for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Maker In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications. An example In the situation of Fig 95, the horizontal direction angle q appears to be somewhere between p and 3p /2 The vertical direction angle f appears to be roughly 1 radian We can t be sure of the exact values of these angles, because we don t have any reference lines to compare them with The radius r is positive, but we have no idea how large it is because there are no radial coordinate increments Another example Imagine that we set the horizontal direction angle q equal to a constant in spherical coordinates For example, let s say that we have the equation Decode Bar Code In .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Code 3/9 Drawer In Visual C#.NET Using Barcode generator for .NET Control to generate, create Code39 image in .NET applications. q = 7p /5 When we work in polar coordinates and set the direction angle equal to a constant, we get a line passing through the origin In spherical coordinates, the horizontal angle in the reference plane is geometrically identical to the polar direction angle Therefore, if k is any realnumber constant, the graph of q=k is a plane that passes through the vertical axis When k = 7p /5, that vertical plane also contains the ray for the direction angle 7p /5, as shown in Fig 96 Code 39 Full ASCII Creator In Visual Studio .NET Using Barcode creation for ASP.NET Control to generate, create Code 39 Full ASCII image in ASP.NET applications. USS Code 39 Generator In VB.NET Using Barcode maker for Visual Studio .NET Control to generate, create USS Code 39 image in .NET applications. Still another example If we set the radius equal to a constant in spherical coordinates, we get the set of all points at some fixed distance from the origin That s a sphere centered at the origin Figure 97 shows what happens when we graph the following equation: Code 3 Of 9 Creation In VS .NET Using Barcode generator for .NET framework Control to generate, create Code39 image in Visual Studio .NET applications. Creating Bar Code In .NET Using Barcode maker for VS .NET Control to generate, create barcode image in Visual Studio .NET applications. r=k in spherical threespace, where k is a nonzero constant
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Figure 97 When we set the radius equal to a constant in spherical coordinates, we get a sphere centered at the origin Spherical Coordinates
Are you confused
Don t get the wrong idea about the meaning of the radius in spherical coordinates It s not the same as the cylindricalcoordinate radius! In spherical coordinates, the radius follows a straightline path from the origin to the point whose coordinates we re interested in This line almost never lies in the reference plane In cylindrical coordinates, the radius goes from the origin to the projection of the point in the reference plane You can see the difference if you compare Fig 91 with Fig 95 Are you still confused
If you ve read a lot of other precalculus texts (and I recommend that you do), you might notice that the order in which we list spherical coordinates is different from the way it s done in some of those other texts You might see the spherical coordinates of a point P go with the radius first, then the horizontal angle, and finally the vertical angle, as P = (r,q,f) Theoretically, it doesn t matter in which order we list the coordinates For any particular values, we re always working with the same point When we want to get from the origin to a point in spherical threespace, most people find it easiest to think of the horizontal angle q first (as in face southeast ), then the vertical angle f (as in fix your gaze at an angle that s p /6 radian from the zenith ), and finally the radius r (as in follow the string for 150 meters to reach the kite ) That s why we use the form P = (q,f,r) Here s a challenge! What sort of graph do we get if we set the vertical angle f equal to a constant in spherical coordinates As an example, draw a diagram showing the graph of the following equation: f = p /4

