It shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola.
Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. In the French engineer Girard Desargues initiated the study of those properties of conics that are invariant under projections see projective geometry.
The line segment joining the vertices of a conic is called the major axis, also called transverse axis in the hyperbola. Post-Greek applications Conic sections found their first practical application outside of optics in when Johannes Kepler derived his first law of planetary motion: Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines.
The line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. A planet travels in an ellipse with the Sun at one focus.
A property that the conic sections share is often presented as the following definition. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane that is its directrix.
A circle is a limiting case and is not defined by a focus and directrix, in the plane however, see the section on the extension to projective planes.
Capital and in St. The focal properties of the ellipse were cited by Anthemius of Tralles, one of the architects for Hagia Sophia Cathedral in Constantinople completed in adas a means of ensuring that an altar could be illuminated by sunlight all day. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola.
Archimedes is said to have used this property to set enemy ships on fire. The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
From the ubiquitous parabolic satellite dish see the figure to the use of ultrasound in lithotripsynew applications for conic sections continue to be found. Parabolic satellite dish antennaSatellite dishes are often shaped like portions of a paraboloid a parabola rotated about its central axis in order to focus transmission signals onto the pickup receiver, or feedhorn.
By a suitable choice of coordinate axes, the equation for any conic can be reduced to one of three simple r forms: For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles and all parabolas have the same shape, only varying in size.
The midpoint of this line segment is called the center of the conic.
A conic is the curve obtained as the intersection of a planecalled the cutting plane, with the surface of a double cone a cone with two nappes. Eighteenth-century architects created a fad for whispering galleries —such as in the U.
Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. A conic section is the locus of all points P whose distance to a fixed point F called the focus of the conic is a constant multiple called the eccentricitye of the distance from P to a fixed line L called the directrix of the conic.
In the remaining case, the figure is a hyperbola. The circle is a special kind of ellipse, although historically it had been considered as a fourth type as it was by Apollonius.
In contrast, ellipses and hyperbolas vary greatly in shape. Definition The black boundaries of the colored regions are conic sections.Learn about the four conic sections and their equations: Circle, Ellipse, Parabola, and Hyperbola.
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse.
Conic section, also called conic, in geometry, any curve produced by the intersection of a plane and a right circular cone.
Depending on the angle of the plane relative to the cone, the intersection is a circle, an ellipse, a hyperbola, or a parabola. Definition of conic section 1: a plane curve, line, pair of intersecting lines, or point that is the intersection of or bounds the intersection of a plane and a cone with two nappes 2: a curve generated by a point which always moves so that the ratio of its distance from a fixed point to its distance from a fixed line is constant.
The pedal curve of a conic section with pedal point at a focus is either a circle or a line. In particular the ellipse pedal curve and hyperbola pedal curve are both circles, while the parabola pedal curve is a line (Hilbert and Cohn-Vossenpp.
). Five points in a plane determine a conic.Download