xy-trace \( (z=0) \): \( x^2 + y^2 = 1 \) (circle radius 1)
Slices at other z: \( x^2 + y^2 = 1 + z^2 \) (circles radius \( \sqrt{1+z^2} \ge 1 \))
Larger circles than xy-trace
\( -\infty < z < \infty \)
xz-trace \( (y=0) \): \( x^2 - z^2 = 1 \) (hyperbola)
intercepts: \( x^2 = 1 \rightarrow x = \pm 1 \) but no z-int
at other y: \( x^2 - z^2 = 1 - y^2 \)
if \( |y| < 1 \) \( (-1 < y < 1) \): \( 1 - y^2 > 0 \)
right side if pos. has x-int but no z-int
if \( |y| > 1 \) then \( 1 - y^2 < 0 \)
rearrange: \( x^2 - z^2 = 1 - y^2 \rightarrow z^2 - x^2 = y^2 - 1 \) (now has z-ints but no x-ints)
yz-trace (\(x=0\)): \(y^2 - z^2 = 1\)
At other \(x\): \(y^2 - z^2 = 1 - x^2\)
At \(x=1\): \(y^2 - z^2 = 0 \rightarrow\) lines (asymptotes of hyperbolas)
Circles get bigger as \(z\) changes.
Like a vase
"Hyperboloid of one sheet"
The diagram illustrates a hyperboloid of one sheet with various cross-sections.
cut y > 1
cut at y < 1 through the "neck"
This page displays a series of six diagrams showing different planar cross-sections of a hyperboloid of one sheet, with the top row showing 3D perspective views and the bottom row showing the corresponding 2D projections.
The equation is given by:
xy-trace (\( z=0 \)):
trace w/ other z:
Circles radius \( \sqrt{z^2 - 1} \), \( |z| > 1 \). Slices are circles above \( z=1 \), below \( z=-1 \), at \( z=\pm 1 \), points.
xz-trace (\( y=0 \)):
hyperbolas w/ z-ints
at other y:
yz-trace (\( x=0 \)):
other x:
same idea, no switching
The following figure illustrates the geometric representation of a hyperboloid of two sheets in a three-dimensional coordinate system.
The general equation for a quadric surface is given by:
cone
hyperbolic paraboloid
