qr code vb.net open source Polar Two-Space in VS .NET

Drawing ANSI/AIM Code 39 in VS .NET Polar Two-Space

Polar Two-Space
Code39 Reader In VS .NET
Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications.
Drawing Code 39 In .NET Framework
Using Barcode maker for .NET Control to generate, create Code 39 Extended image in .NET applications.
Three Basic Graphs
Recognizing Code 39 In .NET
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications.
Make Barcode In Visual Studio .NET
Using Barcode encoder for VS .NET Control to generate, create barcode image in VS .NET applications.
Let s look at the graphs of three generalized equations in polar coordinates In Cartesian coordinates, all equations of these forms produce straight-line graphs Only one of them does it now!
Scan Bar Code In Visual Studio .NET
Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications.
USS Code 39 Printer In C#.NET
Using Barcode maker for Visual Studio .NET Control to generate, create Code 39 image in Visual Studio .NET applications.
Constant angle When we set the direction angle to a numerical constant, we get a simple polar equation of the form
Drawing Code 3/9 In Visual Studio .NET
Using Barcode generator for ASP.NET Control to generate, create Code 3/9 image in ASP.NET applications.
Encode Code-39 In VB.NET
Using Barcode generation for .NET framework Control to generate, create Code-39 image in VS .NET applications.
q=a where a is the constant As we allow the value of r to range over all the real numbers, the graph of any such equation is a straight line passing through the origin, subtending an angle of a with respect to the reference axis Figure 3-2 shows two examples In these cases, the equations are q = p /3 and q = 7p /8
Matrix 2D Barcode Drawer In VS .NET
Using Barcode generator for .NET Control to generate, create Matrix Barcode image in VS .NET applications.
Bar Code Drawer In VS .NET
Using Barcode generator for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications.
p /2
Making Code 128A In Visual Studio .NET
Using Barcode creation for VS .NET Control to generate, create Code 128 Code Set A image in .NET applications.
Generate OneCode In Visual Studio .NET
Using Barcode encoder for VS .NET Control to generate, create Intelligent Mail image in VS .NET applications.
q = p /3
Barcode Drawer In VS .NET
Using Barcode generator for Reporting Service Control to generate, create barcode image in Reporting Service applications.
EAN-13 Creation In Visual Basic .NET
Using Barcode creation for .NET Control to generate, create EAN 13 image in .NET framework applications.
q = 7p/8
Recognize Code 39 In Visual C#.NET
Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Barcode Reader In C#.NET
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.
3p/2
Drawing Barcode In Java
Using Barcode encoder for Java Control to generate, create bar code image in Java applications.
DataMatrix Printer In C#.NET
Using Barcode generator for .NET Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications.
Figure 3-2 When we set the angle constant, the graph is a
Barcode Recognizer In Java
Using Barcode Control SDK for Eclipse BIRT Control to generate, create, read, scan barcode image in BIRT reports applications.
Code 3 Of 9 Maker In Objective-C
Using Barcode printer for iPhone Control to generate, create Code39 image in iPhone applications.
straight line through the origin Here are two examples
Three Basic Graphs
Constant radius Imagine what happens if we set the radius to a numerical constant This gives us a polar equation of the form
r=a where a is the constant The graph is a circle centered at the origin whose radius is a, as shown in Fig 3-3, when we allow the direction angle q to rotate through at least one full turn of 2p If we allow the angle to span the entire set of real numbers, we trace around the circle infinitely many times, but that doesn t change the appearance of the graph
Angle equals radius times positive constant Now let s investigate a more interesting situation Figure 3-4 shows an example of what happens in polar coordinates when we set the radius equal to a positive constant multiple of the angle We get a pair of mirror-image spirals To see how this graph arises, imagine a ray pointing from the origin straight out toward the right along the reference axis (labeled 0) The angle is 0, so the radius is 0 Now suppose the ray starts to rotate counterclockwise, like the sweep on an old-fashioned military radar screen The angle increases positively at a constant rate Therefore, the radius also increases at a constant rate, because the radius is a positive constant multiple of the angle The resulting
p/ 2
3p /2
Figure 3-3 When we set the radius constant, the
graph is a circle centered at the origin In this case, the radius is an arbitrary value a
Polar Two-Space
Positive angle, positive radius
p /2
Negative angle, negative radius
3p /2
Figure 3-4 When we set the radius equal to a positive
constant multiple of the angle, we get a pair of spirals
graph is the solid spiral The pitch (or tightness ) of the spiral depends on the value of the constant a in the equation r = aq Small positive values of a produce tightly curled-up spirals Larger positive values of a produce more loosely pitched spirals Now suppose that the ray starts from the reference axis and rotates clockwise At first, the angle is 0, so the radius is 0 As the ray turns, the angle increases negatively at a constant rate That means the radius increases negatively at a constant rate, too, because we re multiplying the angle by a positive constant We must plot the points in the exact opposite direction from the way the ray points When we do that, we get the dashed spiral in Fig 3-4 The pitch is the same as that of the heavy spiral, because we haven t changed the value of a The entire graph of the equation consists of both spirals together
Angle equals radius times negative constant Figure 3-5 shows an example of what happens in polar coordinates when we set the radius equal to a negative constant multiple of the angle As in the previous case, we get a pair of spirals, but they re upside-down with respect to the case when the constant is positive To see
Three Basic Graphs
Positive angle, negative radius
p /2
3p /2
Negative angle, positive radius
Figure 3-5 When we set the radius equal to a negative
constant multiple of the angle, we get a pair of spirals upside-down relative to those for a positive constant multiple of the angle Illustration for Problem 4
how this works, you can trace around with rotating rays as we did in Fig 3-4 Be careful with the signs and directions! Remember that negative angles go clockwise, and negative radii go in the opposite direction from the way the angle is defined
Copyright © OnBarcode.com . All rights reserved.