## Virtual Laboratory

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## MathKit 6

dynamic mathematics software

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# SAMPLE MODELS

## Conics and Other Remarkable Curves

In this part of our collection we present models of several well-known curves and their constructions.

Conics are the curves obtained by cutting a cone with a plane. Depending on the position of the plane such a curve can be an ellipse, a hyperbola, or a parabola. Alternatively, these curves can be defined in terms of the distances from a point on a curve to two fixed points or, for parabola, to a point and a line. It is the latter definition that we use in the constructions of these curves. Also, we explore the behavior of light rays after reflection in a conic-shaped mirror.

Another kind of curve we consider is a cardioid. This curve is formed by light rays reflected in a circle or in a totally different way, by rolling one circle around another one. ### Ellipse This model illustrates the "focal" definition of an ellipse: an ellipse is the locus of all points in the plane the sum of whose distances to two given points, called foci, is constant.

#### Activity

Construct an ellipse given its two foci and a segment equal to the sum of distances to them from any of its points.

Take a point on the given segment. Draw two circles centered at the foci whose radii are equal to the two parts into which this point divides the segment. Construct the intersection points of the circles and the locus of each of these two points. Together, the two curves thus obtained form an ellipse.

#### Optical Property of an Ellipse After a ray of light issued from an ellipse's focus reflects off the ellipse, it always passes through the other focus. It shows a ray originating from a movable point S and its reflection in the ellipse. The point M at which the ray hits the ellipse can be moved around the ellipse changing the ray's direction. We can see that when S is placed on a focus, the reflected rays converge on the other focus. ### Hyperbola The model illustrates the definition of a hyperbola as the locus of all points in the plane for which the absolute value of the difference of their distances to two given points, called foci, is constant.

#### Activity

Construct a hyperbola given its two foci and a segment equal to the difference of distances to them from any of its points.

Take a point P on the extension of the given segment beyond any of its endpoints. Draw two circles centered at the foci whose radii are equal to the distances from P to the endpoints. Construct the intersection points of the circles and the locus of each of these two points. Each of these loci will consist of two branches, halves of the branches of the hyperbola. Together they form the entire curve.

#### Optical Property of Hyperbola

A ray of light issued from a hyperbola's focus reflects off the hyperbola as if it was issued from the other focus. In other words, the tangent to the hyperbola at its point P bisects the angle F1PF2, where F1 and F2 are the foci. It shows a ray originating from a movable point S and the straight line that extends its reflection in the hyperbola. The point M at which the ray hits the hyperbola can be moved on the hyperbola. We can see that when S is placed on a focus, the extensions of the reflected rays converge on the other focus. ### Parabola A parabola can be defined as the locus of points equidistant from both a given line, called the directrix, and the given point, called the focus.

#### Activity

Construct a parabola given its focus and directrix.

To construct a parabola, take a point P on the directrix, draw the line perpendicular to the directrix at P and the perpendicular bisector to FP (where F is the focus). Then the desired parabola is the locus of the intersection point of the two perpendiculars.

#### Optical Property of Parabola A ray of light issued from a parabola's focus reflects off the parabola in the direction perpendicular to the directrix. It shows a ray originating from a movable point S and its reflection in the hyperbola. The point M at which the ray hits the parabola can be moved along the parabola. We can see that when S is placed on the focus, the reflected rays form a parallel beam perpendicular to the directrix as stated above. ### Cardioid Cardioid as an Epicycloid. When a circle rolls around another, fixed circle, its point traces a curve called epicycloid. The shape of an epicycloid depends on the ratio of the circles' radii. If the radii are equal, the corresponding epicycloid reminds of a heart and for this reason is called a cardioid (from the Greek kardia, "heart").

#### Activity

Construct the cardioid as the locus of a point on a circle that rolls without slipping around a fixed circle of the same radius.

#### Cardioid as a Caustic Curve The caustic of a curve is the envelope of light rays radiating from a point source after reflection in the curve. The caustic of a circle with a light source on the same circle turns out to be a cardioid: you can see a bright heart-shaped trace inside a lighted cup of tea. The model illustrates the caustic of a circle for different positions of the light source.

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