Loci

In geometry, a locus is the set of all possible positions of an object that satisfy some specific condition. For example, you might investigate the locus of points equidistant from two fixed points or the locus of circles that have their center point on a fixed circle and which pass through a fixed point.

Possible paths include straight objects, circles, arcs, polygons, other interiors, and point loci themselves.

In Sketchpad, a locus describes the position of an object while some point (on which the object depends) travels along a path. More formally, a Sketchpad locus is the set of positions of a driven object generated as some driver point—on which the driven object depends—moves to a finite number of positions along a drive path.

A few examples may help: In the first pair of illustrations, point E is the midpoint of segment CD. The top right illustration shows the locus of point E as point C travels along segment AB. The locus of point E forms a smaller segment, parallel to and half the length of segment AB. In this example, point E is the driven object, point C is the driver point, and segment AB is the drive path. In the next pair of illustrations, circle CB is constructed with points C and B defined on circle AB. The locus of circle CB as point C travels along circle AB is the cardiod shown at right. In this example, circle CB is the driven object, point C is the driver point, and circle AB is the drive path.

Driver: Point C
Drive path: Segment AB
Driven object: Point E

Locus of point E

Driver: Point C
Drive path: Circle AB
Driven object: Circle CB

Locus of circle CB

If you don’t find the terms driver, drive path, and driven object helpful in understanding the objects that define a locus, consider using other analogies. Some people prefer to think of a Sketchpad locus as a visualization of an abstract function. In this analogy, an independent variable (the driver) is defined over a particular domain (the drive path). The value of that independent variable (that is, the position of the driver on the drive path) determines the value of some dependent variable (that is, the position of the driven object). Each sample of the locus represents one value of the function, and the entire locus is an approximation of the range of the function. (It’s an approximation because Sketchpad uses only a finite number of ordered pairs—or samples—in constructing the locus.) The abstract function in this analogy is actually the construction by which the driven object relates to the driver.

Others prefer to think of a Sketchpad locus as a more durable form of a traced animation. In that case, you have an animating point (the driver) moving along the path on which it’s constructed (the drive path) that defines some traced object (the driven object). As the animating point moves along its path, the traced object traces out the locus.

The difficulty of naming these dynamic concepts has a long history: When Johan De Witt and Sir Isaac Newton studied locus constructions of the conics in the 17th century, they used the term directrix to refer to what we call the driver. Today, when discussing the same type of locus, mathematicians use directrix to refer to the drive path instead!

Using Plot Properties, you can set the number of samples Sketchpad uses when calculating and displaying a locus.

Mathematically, a locus may describe an infinite number of positions of the driven object. However, to display an infinite number of positions would require a computer to use an infinite amount of time, so Sketchpad instead displays a large (but not infinite) number of possible positions, rather than all possible positions. Each position that Sketchpad does display is called a sample.

Subtopics:
Constructing a Locus
Modifying a Locus