Curves

Polar coordinates. Enter another function in terms of x.

Implicit curves

GeoGebra distinguishes between equations and functions. Functions are written using brackets, as in \(f(x)=x^2\). Equations are written using \(x\) and \(y\), as in \(y=x^2\). In this case \(y\) can be expressed explicitly in terms of \(x\). Since \(y\) is uniquely defined by \(x\), it is a function of \(x\). An equation, however, is a more general concept than a function. Equations describe a relation between \(x\) and \(y\), and it is not always the case that \(y\) can be explicitly written in terms of \(x\). The equation \(x^2+y^2=1\) defines a curve that is the unit circle. All points \((x,y)\) that satisfy the equation, lie on the circle. The curve is implicitly defined by the equation. In GeoGebra such curves are called Implicit curves.

You can always enter a function in either of the two forms, but if you enter a non-polynomial function in equation-form, GeoGebra will change it. Enter y=e^x+sin(x) and y=x^2 in the input bar. Then check out how they are defined in the algebra view!

Commands that take a function as parameter, do not work on equations. If a is the equation y=x^2, you can not use the command TurningPoint on a. In order to use commands for functions, write them as functions!

As long as an equation only involves polynomials in \(x\) and \(y\), GeoGebra can handle them and draw the curve defined by the equation. The equation of an ellipse can be written as \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) and the equation of a hyperbola as \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\). Enter two sliders a and b, then input the equations for an ellipse and hyperbola in the input bar!

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For conic sections it is also possible to use the tools made specifically for conics. These tools can be found under the Ellipse tool.

An example of two animated quartic polynomials in two variables is shown in the applet below.

Animated smiley heart.

Polar coordinates

In GeoGebra you can use polar coordinates when defining a point. When using polar coordinates, a semicolon is used as delimiter, as in A=(r;α). See GeoGebra Tutorial - Polar Coordinates for more information. By letting the angle be represented by a slider t, and by defining a function r(x), it is possible to draw the curve traced out by a point A=(r(t);t). Use the tool Locus, click on the point and then the slider. Under the preferences for Graphics, under the Grid tab, it is possible to show the grid in polar form.

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Polar coordinates can be converted to Cartesian coordinates by letting \(x=r\cos(t)\) and \(y=r\sin(t)\). This can be used for drawing the curves as parametric curves instead of using the Locus tool.

Parametric curves

If the x- and the y-coordinate both depend on a parameter t, they describe a parametric curve with parameter t. Each point on the curve can be described by the coordinates \((x(t),y(t))\). By letting t start at a value start and end at a value stop you can draw the curve k by entering: k = Curve[ x(t), y(t), t, start, stop]

You can always convert a regular function to a parametric function. As an example, the graph of \(f(x)=x^2,-10\leq x \leq 10\) can be drawn as a parametric curve by entering the code:

	k = Curve[t, t^2, t, -10, 10]

By letting a slider represent the stop-value, it is possible to visualize how a graph is drawn. Enter a slider tmax and define the curve:

	k = Curve[t, t^2, t, -10, tmax]

For a more advanced example, see the animated flowers at Home

Make a Lissajous Curve

Animated Lissajous curve. When a=1 it's a circle.

A Lissajous curve is the result of periodic motion along both the x- and the y-axis. Such a curve is shown in the applet above.

The curve in the applet above is made from following command:

	k = Curve[cos(t), sin(a t), t, 0, tmax]

where tmax is the animated slider.

A general Lissajous-curve can be written:

\( \left\{ \begin{align*} x &= A\sin (at+\delta)\\ y &= B\sin (bt) \end{align*} \right. \)

Anaglyphic 3D animation of Lissajous-curve

You need anaglyphic glasses in red and cyan to see the 3D-effect.

Anaglyphic animated Lissajous curve.

Warning: Using anaglyphic glasses when making 3D-worksheets may cause severe headache and dizziness.

Using the beta version GeoGebra 5.0, it is possible to make real 3D curves. Download from following forum post.

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See Damped Lissajous Curves for more information.

Damped Lissjous curve

further info:

Wikipedia: Algebraic curve

Lissajous curves can be drawn by harmonographs

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se