Recipe

Identify and sketch a quadric surface from its equation

Use when: you have an equation in x, y, z with squared terms and need to name which of the six quadric families it belongs to, or recognize it as a cylinder instead

Goal: the surface's family name, plus enough of its cross-sections to sketch it

Taught in Cylinders and quadric surfaces

1. 1

   Move terms so the equation reads as one of the recognizable patterns: squared terms summed against a constant, or squared terms summed against a single isolated linear variable.

   Example

   x^2+y^2-z^2=1 is already in "sum of squares equals a constant" form.

   Why: the six families are told apart by exactly this arrangement, how many variables are squared, what sign each one carries, and which side of the equation the constant or the lone linear variable sits on.

2. 2

   Scan the standard-form equation from Step 1 for a variable that never appears at all, not even squared.

   • exactly one variable is missing from the equation — the surface is a cylinder over whatever 2D curve the other two variables trace, with the missing variable's axis left free; stop here.
   • all three variables appear — continue to Step 3 to sort among the quadric families.

   Why: a cylinder's whole identity is what it's missing, not a squared-term sign pattern, so ruling this out first avoids force-fitting a cylinder into one of the six families below.

3. 3

   Note how many squared terms are positive versus negative, and whether the equation equals a nonzero constant, has one variable alone and unsquared on the other side, or equals zero.

   Example

   x^2+y^2-z^2=1: two positive squared terms, one negative, equal to a nonzero constant.

   • equals a nonzero constant — ellipsoid or hyperboloid family; continue to Step 4.
   • one variable stands alone, unsquared, on the other side — paraboloid family; continue to Step 4.
   • equals zero, with mixed signs among the squared terms — cone, apex at the origin.

   Why: this narrows the search from six families down to two or three before a single trace is even drawn.

4. 4

   Freeze each variable in turn, setting x=0, then y=0, then z=0, and identify the 2D curve each cross-section draws.

   Example

   x^2+y^2-z^2=1: the z=0 trace is x^2+y^2=1, a circle; the y=0 trace is x^2-z^2=1, a hyperbola; the x=0 trace is y^2-z^2=1, also a hyperbola. Two hyperbolas plus one ellipse, with a nonzero constant, matches the hyperboloid pattern.

   • all three traces are ellipses — ellipsoid.
   • two traces are hyperbolas and one is an ellipse, with a nonzero constant — hyperboloid.
   • two traces are parabolas opening the same way and one is an ellipse — elliptic paraboloid.
   • two traces are parabolas opening opposite ways and one is a hyperbola — hyperbolic paraboloid, the saddle.

   Why: the trace test turns an abstract sign pattern into an actual picture, three 2D curves that pin down the family unambiguously.

5. 5

   Look at which side of the equation the positive constant actually sits on.

   • the two positive squared terms' side already equals the constant, as in x^2+y^2-z^2=1 — one sheet, a single connected tube.
   • the lone negative term's variable, once isolated, equals the constant, as in z^2-x^2-y^2=1 — two sheets, a pair of separate bowls.

   Why: whichever variable ends up isolated with the constant sets the surface's axis. Keeping the constant on the side with two positive squares leaves every cross-section nonempty, while isolating the other variable instead forces that variable's magnitude past a minimum, splitting the surface into two disconnected pieces.