Curve

In order to produce a curve using an apparatus, drag the gray triangle point P with the mouse.  This will cause another black round point to move (among other points).  This point is usually unlabeled or labeled with an  X.  As it moves, it traces out a curve.

Setting up an apparatus.  If necessary get a new window with the New Window command (Command-n) under the Windows menu.  Then under the Apparatus menu select the apparatus desired.

Operating an apparatus.  Holding down the Command key when clicking on  P  constrains  P  to move only in the horizontal or vertical directions.  Some apparati may override this and constrain the movement of  P  themselves.  Holding down the Shift key when clicking on  P  prevents any constructed curves from being erased.  In this way, one can produced a family of curves as parameters are varied.

Synthetic Apparati

In synthetic apparati, the curve is usually determined by its relation to other points in the plane.  Some of these points, usually black round ones, can be dragged to new locations.  This will determine a new curve.  Some of the points, usually gray round ones, are determined by the black round ones.  They cannot be dragged.  Often they move when  P  does.

The basic idea behind a synthetic appratus is that the relation between the point  X  on the curve and the point  P  is shown by lines, circles, etc. connecting the points.
By moving  P  and by careful observation, one sees in the changing lines, circles, etc. how the curve is constructed.
This idea is based on what is called a `linkage.'  A linkage is an apparatus consisting of rods linked together in various ways.  At a connection of two rods, the rods may be free to rotate, one may be able to slide, or perhaps both can slide.  (See Yates, Curves and Their Properties, published by the National Council of Teachers of Mathematics, for pictures of linkages.)  Another kind of synthetic apparatus is the familiar toy Spirograph.  In this kind of apparatus, circles roll on circles.

Some of the apparati have Inspectors.  These inspectors allow for a more analytic output and/or input.  They might display the coordinates of points and allow them to be altered.  Other properties of the curve might be displayed such as its equation.

Analytic Apparati

In analytic apparati, the curve is usually determined by an algebraic formula for a function.  These apparati require the use of the Inspector and usually display it automatically.  The function(s) defining the curve may be entered through the Inspector.  Three different ways of describing a curve are now used.

Cartesian coordinates. One describes the  y-coordinate as a function of  the  x coordinate of  P.  As P is dragged a vertical line passing through it moves with it.  This line intersects the curve at the point given by the function.  

Polar coordinates. In polar form one describes the radius (distance from the origin) as a function of the angle t of the segment from the origin to the point  P and the positive x-axis.  As P is dragged, the point on the curve appears on the radial line through the origin and P.

Parametrization of cartesian coordinates.  Here the x- and y-coordinates a defined by functions of an independent variable t.   The value of   t  is determined by a parametrizer supplied by the apparatus.  This is often a line segment representing an interval of  t  values. Other common features usually include being able to set the coordinates of the intersection of cross-hairs called the origin or mid-point of the window and the scaling factor.

Inspectors

An inspector can be used to inspect the parameters used to construct the curve.  Sometimes other information is given too.  One caution:  when you modify a parameter with an inspector, sometimes there is no immediate apparent change in the curve.  You might want to construct various curves using the Shift key while you vary a parameter.  The apparatus will leave whatever curves that have been constructed on the screen.  An inspector may also offer some predefined examples for the user to investigate.  The analytic apparati do this.

With synthetic apparati an inspector gives the user a more analytic description of the curve.  Coordinates and equations may be displayed and changed.  With analytic apparati an inspector is the only way an equation may be entered.

More Help

Some apparati have their own specific help messages.  One may view them by utilizing the pop-up menu on this help panel.

Other Features

*	User can copy apparatus and constructed curve to pasteboard.  This can be pasted into another document.
*	User can program his/her own apparatus and have it dynamically loaded. (Not in this version.  If you are interested in such a version, contact Michael Rogers at mkr@cs.amherst.edu.)
*	Examples are defined by string tables: the user can alter or extend the string files to suit.  See Curve.app/Strings/*Example.strings.