Conic Sections Introduction
Friday, 13 December 2024 12:15 pm
The Greek mathematician Menaechmus, studied curves formed by the intersecting of a perpendicular plane and a cone
two millennia ago. He found that the intersection would form curves, which now became known as conic sections.
Apollonius of Perga also studied the curves formed by the intersection of a plane and a double right circular cone. He
discovered many properties of these curves and coined the following terms: ellipse, parabola, and hyperbola.
The cone was thought to have two parts that extended infinitely in both directions. A line lying entirely on the cone is
referred to as a generator and all generators of a cone pass through the intersection of the two parts called a vertex.
A conic section (or simply conic) is therefore a curve
formed by the intersection of a plane and a double right
circular cone.
Ellipse - if the cutting plane is not parallel to any
generator.
Circle - If the cutting plane is not parallel to any generator
but is perpendicular to the axis.
Hyperbola - if the cutting plane is parallel to two
generators.
Parabola - if the cutting plane is parallel to one and only
one generator.
Picture on page "Conic Sections Introduction"
The conic section above are called non-degenerate conics. It is when the cutting plane does not
pass through the vertex of the cone. When the cutting plane intersects the vertex of the cone, it is called
a degenerate cone. Some examples are: a point, line, and two intersecting lines.
If the early mathematicians were worried primarily about geometrical properties of conics, other
mathematicians discuss conics algebraically as curves of second degree equations rather than as
sections of the cone. English mathematician, John Wallis (1616-1703) was one of the first to describe
that all conics can be written in the form:
=0
References:
Argel, A., & Mallari, M.T., (2022). Next Century Mathematics 2nd Edition. Phoenix Publishing House, Inc.
Pre-Calculus 11 Page 1
Friday, 13 December 2024 12:15 pm
The Greek mathematician Menaechmus, studied curves formed by the intersecting of a perpendicular plane and a cone
two millennia ago. He found that the intersection would form curves, which now became known as conic sections.
Apollonius of Perga also studied the curves formed by the intersection of a plane and a double right circular cone. He
discovered many properties of these curves and coined the following terms: ellipse, parabola, and hyperbola.
The cone was thought to have two parts that extended infinitely in both directions. A line lying entirely on the cone is
referred to as a generator and all generators of a cone pass through the intersection of the two parts called a vertex.
A conic section (or simply conic) is therefore a curve
formed by the intersection of a plane and a double right
circular cone.
Ellipse - if the cutting plane is not parallel to any
generator.
Circle - If the cutting plane is not parallel to any generator
but is perpendicular to the axis.
Hyperbola - if the cutting plane is parallel to two
generators.
Parabola - if the cutting plane is parallel to one and only
one generator.
Picture on page "Conic Sections Introduction"
The conic section above are called non-degenerate conics. It is when the cutting plane does not
pass through the vertex of the cone. When the cutting plane intersects the vertex of the cone, it is called
a degenerate cone. Some examples are: a point, line, and two intersecting lines.
If the early mathematicians were worried primarily about geometrical properties of conics, other
mathematicians discuss conics algebraically as curves of second degree equations rather than as
sections of the cone. English mathematician, John Wallis (1616-1703) was one of the first to describe
that all conics can be written in the form:
=0
References:
Argel, A., & Mallari, M.T., (2022). Next Century Mathematics 2nd Edition. Phoenix Publishing House, Inc.
Pre-Calculus 11 Page 1
