Secciones Cónicas
A collection of several 2D and 3D GeoGebra applets for studying the conics (ellipse, parabola, and hyperbola)
Table of Contents
- Secciones del cono
- Conic Sections
- Conic Section - A Geometric Construction using Eccentricity
- Locus of the centers of all circles tangent to two circles
- Some fascinating properties of the conic sections
- Dandelin Spheres
- Elipse
- On the geometric definition of ellipse
- Reflective Properties of the Conics
- Elipse: Reflective Property
- Reflexiones repetidas en la elipse
- Hipérbola
- How to construct a hyperbola
- Hipérbola: método del jardinero y elementos formales
- Parábola
- How to construct a parabola
Conic Sections
A conic section is the intersection of a plane and a cone. The three conic sections ellipse, parabola or hyperbola can be produced by changing the slope of the plane (that is, the angle between the axes of the cone and the intersecting plane).[br]In the applet below we consider an infinite cone with an angle [math]\alpha[/math].[br][*]Play the animation to see a demonstration of the conics, hyperbola, parabola, ellipse and circle.[br][*]Stop the animation and explore additional special cases by changing the angle of the plane (drag the orange point) and the location of the plane (drag the brown point).
On the geometric definition of ellipse
The applet demonstrates the following:[br][i]An ellipse is the set of all points in the plane, the sum of whose distances to two fixed points (foci) remains constant.[/i][br][list][br][*]Select the length of a piece of string by dragging the endpoints of the blue segment.[br][*]Drag the [b]orange[/b] point to select the position of the focus F1 along [b]the x-Axis or the y-Axis[/b]. The other focus F2 is symmetrical to F1 with respect to the origin.[br][/list][br]A string with the selected length is attached to both foci and is kept tight by the tip of the pencil.[br][list][br][*]Drag the tip of the pencil or press the “Draw” button to trace all points on the plane that satisfy the above definition.[br][*]Hide the pencil by pressing the "Pencil ON/OFF" button; show the ellipse by pressing “Show Ellipse” button, and explore the curve by changing the positions of the foci and the length of the constructing string. [br][*]Bring the two foci to the origin to see the circle as a special case of the ellipse.[br][*]Click on “Labels” to see some terminology.[br][/list]
On the geometric definition of ellipse
How to construct a hyperbola
A hyperbola is the set of all points in the plane for which the absolute value of the difference of the distances to two fixed points [math] F_1[/math] and [math]F_2[/math] (the foci) is a constant. This constant equals the distance between the vertices of the hyperbola.[br]In the applet below points [math]L[/math], [math]M[/math] and [math]N[/math] are points on one line, so [math]| NL-NM|=LM.[/math][br][*]Drag point [math]N[/math] along the line or use the animation button to construct points from the hyperbola.[br][*]Click the checkbox to see the construction.[br][*]Drag [math]L[/math] or [math]M[/math]. What happens when [math]LM>F_1F_2[/math]?
How to construct a hyperbola
How to construct a parabola
Given is a line [math] d [/math] and a point [math] F[/math], not on [math]d[/math]. A parabola is the set of all points in the plane that are equidistant from [math]F[/math] (the focus) and [math]d[/math] (the directrix).[br][list][br][*]Drag the blue point M along the directrix to create points from the parabola.[br][*]Click the checkbox to see the construction.[br][*]Drag the Focus and the Directrix to change the parabola.[br][/list]