Help with cones

[chyo]

Hi, I'm a new member here and firstly I wish to briefly share what had happened to me when I registered for an account here a few days ago: I had posted a new thread but a few hours after that it disappeared, together with my account! :eek: The email address I used was not even registered with the forum! freaky lol, but anyway here i am again with a new account and i hope i can get some help from the community here with my physics and math doubts...

ok so the question i had initially posted with my nonexistent account was, what is the general cartesian equation of a right circular cone? Given that the origin is at the centre of the base, the positive z-axis is towards the apex, and the height is h and radius of base is r. The equation that I get is (h-z)^2 = (h/r)^2 (x^2+y^2). Can anyone confirm this?

Also, what significance does it mean when the equation of a cone becomes ax^2 + by^2 = (h-cz)^2? Does this automatically 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?

[HallsofIvy]

Hi, I'm a new member here and firstly I wish to briefly share what had happened to me when I registered for an account here a few days ago: I had posted a new thread but a few hours after that it disappeared, together with my account! :eek: The email address I used was not even registered with the forum! freaky lol, but anyway here i am again with a new account and i hope i can get some help from the community here with my physics and math doubts...

ok so the question i had initially posted with my nonexistent account was, what is the general cartesian equation of a right circular cone? Given that the origin is at the centre of the base, the positive z-axis is towards the apex, and the height is h and radius of base is r. The equation that I get is (h-z)^2 = (h/r)^2 (x^2+y^2). Can anyone confirm this?
Notice what happens if you look only at the xz-plane or yz-plane. Letting y= 0 gives (h-z)^2= (h/r)^2(x^2) or (h-z)= +/- (h/r)x, two lines that intersect at (0, h). If you let x= 0 you get a similar thing in the yz-plane. Those lines are, of course, where the cone intersects the plane. Yes, that is the equation of the cone.

Also, what significance does it mean when the equation of a cone becomes ax^2 + by^2 = (h-cz)^2? Does this automatically 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?
Are you sure you don't have that backwards? Adding MORE independent parameters makes it more general not less!
Again, look what happens if you take y= 0. You get ax^2= (h-cz)^2 or
sqrt(a)x= h-cz. That's a line that goes through (0,0,h/c) and (h/sqrt(a),0,0).
If you let x= 0, you get by^2= (h-cz)^2 or sqrt(b)y= h-cz. That's a line that goes through (0,0, h/c) and (0,h/sqrt(b),0). Yes, it is a cone with vertex at (0,0,h/c) but is is not true that "the height of the cone is equal to its base radius"- in fact, there is no "base radius"; the base of the cone is no longer a circle, it is an ellipse with semi-axis length in the x-direction h/sqrt(a) and in the y-direction h/sqrt(b).