The graphics below are generated with Graphics Explorer, my equations grapher program.download Graphics Explorer Theory. He is famous for his research on waves and oscillations. A particular kind of graphs are called "Lissajous curves", 3 examples are pictured right. Lissajous curves are made by so called "parametric functions".
"common" functions, like y = 5sin(x) , produce a -single- y value for a value of x. So, it is not possible to make graphs of spirals or even a circle, which needs 2 -y- values for each value of x. Parametric functions overcome this limitation by the following tric: instead of y = f(x) we write: y = g(v) and x = h(v) so x and y are both functions of a new variable v. (Graphics-Explorer uses v, often this variable is called t) An example: if y = 5sin(v) , x = 5cos(v) and v has domain 0..6.28 (2*p radians), the plotted curve is a circle with centre (0,0) and a radius of 5. In general:
Molested Lissajous curves. Lissajous curves, like common functions, will be smooth, without sharp angles. The picture right is the result of the steps of v being too large. This is how GraphicsExplorer paints parametric functions:
If the steps are too large, straight lines with angles instead of smooth curves will result. Practice. Start program Graphics-Explorer or goto [ Graphics-Explorer ] page first.(download program by clicking on download (lightning) logo at top of page). Select AUTOPLOT and REPLACE modes. Type : y = 5sin(v) ; x = 5cos(v) and press ENTER.Set start = 0 , end = 10 en steps = 100. Press ENTER A circle with centre (0,0) and radius of 5 is plotted. Erase graphics by menu:general:restart. Type : y = 5sin(a*v) ; x = 5cos(b*v) Type ENTER or click PLOT button. Move mousepointer over constants A and B and press (and hold) left or right mousebutton. As the constants change, the function will be repaint, using the new values. More accurate variations are obtained by decreasing the constants (+/-) increment value. A higher number of steps for v will produce smoother curves. Try this: y = c*sin(a*v) ; x = c*cos(b*v) Constant c is de size of the graphic. Other Curves. Interesting curves and artwork can be created by combining movements.Type:
y = 5sin(a*v) + c*sin(b*v) ; x = 5cos(a*v) + c*cos(b*v)
A point rotates over a circle with radius = 5. The speed is a (radians/v). This point is the centre of a circle with a radius of c. The pen moves over this second circle with a speed of b. Some examples:
Another example. Type: y = c*sin(a*v)(1 + sin(b*v)) ; x = c*cos(a*v)(1 + sin(b*v)) This represents a point moving over a circle with a speed of a (radians/v). The radius of the circle is modulated by the factor (1 + sin(b*v)).
Also, graphics of different formula's and colors can be combined. Use menu:graphics:color to select other colors for background, grid or curves. Success and have fun! Note Design artist Jennifer Townley builds machines that draw Lissajous curves.Visit her website [HERE] |
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