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The general equation for this type of paraboloid is x2/a2 + y2/b2 = z. If a = b, intersections of the surface with planes parallel to and above the XY plane produce circles, and the figure generated is the paraboloid of revolution.
It has a distinctive nose-cone appearance. This surface is called an elliptic paraboloid because the vertical cross-sections are all parabolas, while the horizontal cross-sections are ellipses.
The hyperbolic paraboloid is a doubly ruled surface, and thus can be used to construct a saddle roof from straight beams.
There are six different quadratic surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the double cone, and hyperboloids of one sheet and two sheets. Rather than memorize the equations, you should learn how to examine cross-sections to figure out what surface a given equation represents.
y=ax2+bx+c. In this equation, the vertex of the parabola is the point (h, k). You can see how this relates to the standard equation by multiplying it out: y=a(oh)(oh)+KY=ax22ahx+ah2+k.
The paraboloid of revolution is the surface obtained by the revolution of a parabola around its axis.
A hyperbolic paraboloid is an infinite surface in three dimensions with hyperbolic and parabolic cross-sections. A couple of ways to parameterize it and write an equation are as follows: z = x2 — y2. Or. x = y z.
Quadratic Surfaces. Ax2+By2+Cz2+Day+Ex+Fez+Go+HY+Jr+K=0. When a quadratic surface intersects a coordinate plane, the trace is a conic section. An ellipsoid is a surface described by an equation of the form x2a2+y2b2+z2c2=1.
Math 2163. p.1/9-Page 2 Cylinders A cylinder is a surface that consists of all lines (rulings) that are parallel to a given line and pass through a given plane curve. A quadratic surface is the graph of a second-degree equation in three variables x, y and z.
[2] classifies quadratic surfaces in the following way : Rank of ME. Signature of ME. Surface. (3, 0, 0) or (0, 3, 0) Ellipsoid. ... (2, 1, 0) or (1, 2, 0) 1 or 2-sheeted hyperboloid or cone. (2, 0, 1) or (0, 2, 1) Elliptic paraboloid or elliptic cylinder. ... (1, 1, 1) Hyperbolic paraboloid or hyperbolic cylinder.
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