Axis of Conic Dimension

You must have heard of the Euclidean plane; it has various properties which can be used in varied definitions. One such property defines the non-circular conic. Similarly, there are different types of conics; the type is determined by the value of eccentricity.

What are conic sections?

Conic sections are the nondegenerate curves produced by the intersection of a plane with one or two nappes (thick sheet-like) of a cone.

• For a plane perpendicular to the axis of the cone, circle is produced
• For a plane that is not perpendicular and intersects only single nappe, 
• the curve is either ellipse or a parabola
• Plane intersecting both nappes the curve produced is hyperbola.
• The ellipse and hyperbola are called central conics 

These are the main three types of conics: ellipses, parabola and hyperbola. There is also, a fourth of which is not very common, Apollonius.

Before getting in detail about conic section there are a few term that need to be familiarized first ,which are;

Conic Parameters:

• Eccentricity(e): The eccentricity is the distance from any point to its focus, divided by the perpendicular distance from that point to its nearest directrix. The eccentricity value is always a constant for any conics.
• FOCUS: Rays reflected from the curve converge at a single point this is the focus of the given conic.as parabola has one focus while ellipse has 2
• DIRECTRIX(r): It is considered as the locus of all points or we can say that it is the line together with the point.It is same with the case of directrix a hyperbola has two directrix and a parabola has only one directrix

What is the Axis of Conic Dimension ?

• Principle axis: It is the line joining the foci of ellipse and hyperbola, and midpoint the curve’s center.
• Major axis: The chord between the two vertices: the longest chord of an ellipse, the shortest chord between the branches of a hyperbola. Its half-length is the semi-major axis (a). When an ellipse or hyperbola are in standard position as in the equations below, with foci on the x-axis and center at the origin.
• Minor axis: The shortest diameter of an ellipse, and its half-length is the semi-minor axis (b), the same value b as in the equation below. In case of a hyperbola the parameter b in the standard equation is also called the semi-minor axis.
• Equations;
• l=pe
• c=ae
• p+e=a ⁄ e

Cartesian coordinates of different conics;

• Circle: x² + y² = a²
• Ellipse: x²/a² + y²/b² = 1
• Parabola: y² = 4ax with a > 0
• Hyperbola: x²/a² − y²/b² = 1
• Rectangular hyperbola: xy = c²/2

CONCLUSION

So, we can conclude by saying that conics define most of Euclidean geometry, it mainly focuses on the conical sections produced during different intersections which are an ellipse, hyperbola, and parabola. We have discussed their peculiarities and properties along with the main parameters required in order to understand and derive the equations. Conics is an important category in mathematics as it explains a lot about structure and position of curved design in space. This property is implied in real life too i.e., in constructional and architectural purposes and other research purposes.