Need help on understanding cone geometry

[chyo]

Hi all,

Homework Statement

Given a right circular cone with origin at the centre of the base, the positive z-axis pointing towards the apex, and the height is h and radius of base is r. What is the cartesian equation of the cone?

Homework Equations

The Attempt at a Solution

The equation that I get is (h-z)^2 = (h/r)^2 (x^2+y^2). Can anyone confirm this?

Assuming that my above equation is correct, how is it that the general equation of a cone is instead x^2 + y^2 = z^2? Where did the extra terms from the first equation go to?

Also, what significance does it bring when the equation of a cone becomes ax^2 + by^2 = (h-cz)^2? If I compare it with the equation i obtained, I suppose that this should mean that the height of the cone is equal to its base radius? What about the constants a, b, c; what do they represent in the physical sense?

Thanks much!

 

[Dick]

Your equation looks fine to me. x^2 + y^2 = z^2 is the equation a 45 degree cone with apex at the origin. Not very general. And for this one, ax^2 + by^2 = (h-cz)^2, if a!=b, that's a vertical cone (axis along z) with an elliptical cross-section.

 

[Architeuthis Dux]

Hello,

The cartesian equation of a cone can be written as x^2 + y^2 = (z^2 / (h/r)^2). This equation represents all points on the surface of the cone, where the distance from the origin to any point (x, y, z) on the surface is equal to the height of the cone (h) divided by the base radius (r). This can also be written as (h-z)^2 = (h/r)^2 (x^2+y^2), which is equivalent to the equation you obtained.

The equation x^2 + y^2 = z^2 is a special case of the general equation, where the height of the cone (h) is equal to the base radius (r). In this case, the cone is a right circular cone, meaning that the apex is directly above the center of the base and the angle between the slant height and the base is 90 degrees. This is the most common type of cone that is studied in math and science.

The equation ax^2 + by^2 = (h-cz)^2 represents a general conic section, which includes not only cones but also other shapes such as circles, ellipses, and hyperbolas. The constants a, b, and c represent the shape and orientation of the conic section. For a cone, a and b are equal to 1, indicating that the cross-section of the cone is circular. The constant c represents the angle of the cone, with a larger value indicating a steeper angle.

I hope this helps to clarify your understanding of cone geometry. Let me know if you have any other questions.