C Programming

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There are several options to draw a circle using C or C++ without the built-in functions...

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If you can draw a single pixel at a time, use a for loop to iterate Angle from 0 to 2*pi. Then plot a pixel with x=cos(Angle) and y=sin(Angle). Smaller steps will give better results. Lines from one point to the next may look nicer.

Generally, you can use a line from one pixel to the next to draw a polygon. The number of sides is determined by the number of steps from 0 to 2*pi. A circle is a polygon with an "infinite" number of sides (compared to the display resolution). Experiment with non-integral steps to see that the polygon can end at any place. Use an integral number of steps to end where the circle began. Start and stop at fractions of 2*pi, or step backwards, to create arcs. Speed up the code by using lookup tables to calculate sin() and cos(). Hint: they can be the same table, and don't need to be a complete 2*pi circle.

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You can also draw the circle without the sin() and cos() functions if you understand the trigonometry behind sin() and cos(). Recall that sin(theta) is radius / y, and cos(theta) is radius / x, given that x and y are the two sides of a right triangle and that radius is the hypotenuse.

By Pythagoream's theorem, x2 + y2 is radius2. It is then simple to solve for x or y, given the other along with radius. You also do not need to compute for the whole circle - you can compute for one quadrant, and generate the other three quadrants by symmetry. You generation loop would, for example, simply iterate from origin to radius as x by delta x, generating y, and reflecting that in the other three quadrants. You can also compute for one half of a quadrant, and use both symmetry and reflection to generate the other seven half quadrants.

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0SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.SUM is a built-in function with Excel that has been specially designed to sum a range of cells. It just needs to know what the cells are and then it can add them all up. The method of doing that has been programmed into Excel by its developers. This leaves us with the simple task of using the function without needing to know the specifics of exactly how it does it.

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