Elementary Surfaces: Spheres, Cylinders, Cones, Conoids and Quadrics

1. Spheres

1° Sphere The sphere with centre \(C(a, b, c)\) and radius \(R\): \[ (x - a)^2 + (y - b)^2 + (z - c)^2 = R^2. \]

This is the definition "all points at distance \(R\) from \(C\)", with the distance formula squared. Expanding gives an equation of the form

\[ x^2 + y^2 + z^2 + 2Ex + 2Fy + 2Gz + H = 0, \]

recognizable by two features: the squares have equal coefficients, and there are no mixed terms \(xy, yz, zx\). Conversely, such an equation can be brought back to the centre–radius form by completing squares: the centre is \((-E, -F, -G)\) and \( R^2 = E^2 + F^2 + G^2 - H \). If this number is negative, there are no real points (an "imaginary sphere"); if it is zero, only the centre.

Example 1 \( x^2 + y^2 + z^2 - 2x + 4y - 6z - 2 = 0 \). Completing squares: \[ (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 2 + 1 + 4 + 9 = 16: \] the sphere with centre \((1, -2, 3)\) and radius \(4\).

2. The method: describe the motion, then eliminate

Cylinders, cones and conoids are ruled surfaces: they are swept out by a moving straight line, the generatrix (generator), which slides along a guiding curve \((c)\), the directrix. In this chapter the directrix is given as the intersection of two surfaces:

\[ (c):\quad F_1(x, y, z) = 0, \qquad F_2(x, y, z) = 0. \]

The recipe is the same in all three cases:

Let \(M_1(x_1, y_1, z_1)\) be the point where the generatrix meets the directrix. It satisfies \( F_1(x_1, y_1, z_1) = 0 \), \( F_2(x_1, y_1, z_1) = 0 \).
Write the equations of the generatrix through \(M_1\): these connect \((x, y, z)\), a general point of the generatrix, with \((x_1, y_1, z_1)\).
That gives four equations. Eliminate \(x_1, y_1, z_1\): what remains is one equation \(F(x, y, z) = 0\), the surface.

The logic: a point \(M(x, y, z)\) is on the surface exactly when some point \(M_1\) of the directrix exists that puts \(M\) on a generatrix. Eliminating \(M_1\) removes the "some" and leaves a condition on \(M\) alone. In practice the easiest way to eliminate is usually to express \(x_1, y_1, z_1\) through \(x, y, z\) from the generatrix equations and substitute into the directrix.

3. Cylindrical surfaces

2° Cylindrical surface Given a curve \((c)\) and a vector \( \vec p = \{l, m, n\} \), the line that slides along \((c)\) staying parallel to \(\vec p\) sweeps out a cylindrical surface. If the generatrix meets \((c)\) at \(M_1(x_1, y_1, z_1)\), then \[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}; \qquad F_1(x_1, y_1, z_1) = 0; \qquad F_2(x_1, y_1, z_1) = 0. \] Eliminating \(x_1, y_1, z_1\) from these four equations gives one relation \(F(x, y, z) = 0\): the equation of the cylindrical surface.

Figure 1. Left: a cylindrical surface, swept by a line sliding along the directrix \((c)\) (red) while staying parallel to \(\vec p\). Right: a conical surface, swept by a line through the fixed vertex \(P\) and a moving point \(M_1\) of \((c)\).

Special cases An equation missing one variable is a cylinder whose generatrices are parallel to that coordinate axis: \( F(x, y) = 0 \) (generatrices \(\parallel Oz\)), \( F(x, z) = 0 \) (\(\parallel Oy\)), \( F(y, z) = 0 \) (\(\parallel Ox\)).

Why: if \(z\) does not appear, then whenever \((x, y, z_0)\) satisfies the equation, so does \((x, y, z)\) for every \(z\): together with each point, the whole vertical line through it belongs to the surface. So in space, \(x^2 + y^2 = 1\) is not a circle but a circular cylinder, and \(y = x^2\) is a parabolic cylinder. This is a common trap: always ask in which space an equation lives.

Example 2 · An oblique cylinder Directrix: the circle \( x^2 + y^2 = 1,\ z = 0 \); generatrices parallel to \( \vec p = \{1, 0, 1\} \).
Since \(z_1 = 0\), the generatrix through \(M_1\) is \( \frac{x - x_1}{1} = \frac{y - y_1}{0} = \frac{z}{1} \), so \( x_1 = x - z \), \( y_1 = y \). Substituting into \( x_1^2 + y_1^2 = 1 \): \[ (x - z)^2 + y^2 = 1. \] Each horizontal section \(z = h\) is the unit circle shifted by \(h\) in the \(x\)-direction: a leaning cylinder.

4. Conical surfaces

3° Conical surface Given a curve \((c)\) and a point \(P(a, b, c)\), the line through \(P\) that slides along \((c)\) sweeps out a conical surface with vertex \(P\). If the generatrix meets \((c)\) at \(M_1(x_1, y_1, z_1)\), it is the line through \(P\) and \(M_1\): \[ \frac{x - a}{x_1 - a} = \frac{y - b}{y_1 - b} = \frac{z - c}{z_1 - c}, \qquad F_1(x_1, y_1, z_1) = 0, \qquad F_2(x_1, y_1, z_1) = 0. \] Eliminating \(x_1, y_1, z_1\) gives the equation \(F(x, y, z) = 0\) of the conical surface.

(Here \(c\) in \(P(a, b, c)\) is a coordinate, not the curve \((c)\); the book uses the same letter for both.)

Example 3 · The circular cone Vertex at the origin, directrix the circle \( x^2 + y^2 = 1,\ z = 1 \). The generatrix through \(O\) and \(M_1(x_1, y_1, 1)\): \( \frac{x}{x_1} = \frac{y}{y_1} = \frac{z}{1} \), so \( x_1 = \frac xz \), \( y_1 = \frac yz \). Substituting: \[ \frac{x^2}{z^2} + \frac{y^2}{z^2} = 1 \quad\Longrightarrow\quad x^2 + y^2 = z^2. \] The double cone with half-angle \(45^\circ\), both nappes (the generatrices are whole lines, so they continue through the vertex).

Recognizing cones: a cone with vertex at the origin has a homogeneous equation: replacing \((x, y, z)\) by \((tx, ty, tz)\) multiplies it by a power of \(t\). That is the algebraic form of "with every point, the whole line through the origin".

5. Conoids

4° Conoid Given a curve \((c)\), a line \(p\) and a plane \(\Pi\), the set of lines that are parallel to the plane \(\Pi\) and pass through points of both the line \(p\) and the curve \((c)\) is called a conoid.

Think of a spiral staircase: every step is horizontal (parallel to the floor \(\Pi\)), starts at the central pole (the line \(p\)) and ends on the outer railing (the curve \((c)\)). The helicoid, the surface of such a staircase, is the classic conoid.

Equation of a conoid Let \(p\) be perpendicular to \(\Pi\), and choose the coordinates so that \(p\) is the \(z\)-axis and \(\Pi\) is the plane \(z = 0\). A generatrix is then a horizontal line meeting the \(z\)-axis: \[ y = ax, \qquad z = b. \] For its intersection \(M_1(x_1, y_1, z_1)\) with \((c)\): \[ F_1(x_1, y_1, z_1) = 0, \quad F_2(x_1, y_1, z_1) = 0, \quad y_1 = ax_1, \quad z_1 = b. \] Eliminating \(x_1, y_1, z_1\) gives a relation \(\varphi(a, b) = 0\), and then \[ \varphi\Big(\frac yx, z\Big) = 0 \] is the equation of the conoid.

The last step uses that every point \((x, y, z)\) of the generatrix \(y = ax, z = b\) has \( a = \frac yx \), \( b = z \). The parameters \(a\) (the slope of the "step") and \(b\) (its height) must be related for the step to reach the curve \((c)\), and \(\varphi(a, b) = 0\) is exactly that relation.

Example 4 Directrix: the circle \( y^2 + z^2 = 1 \) in the plane \( x = 1 \). With \(x_1 = 1\), \(y_1 = a\), \(z_1 = b\), the circle gives \( \varphi(a, b) = a^2 + b^2 - 1 = 0 \). The conoid: \[ \Big(\frac yx\Big)^2 + z^2 = 1 \quad\Longleftrightarrow\quad y^2 = x^2(1 - z^2). \] At height \(z = b\), the section is the pair of lines \( y = \pm\sqrt{1 - b^2}\,x \): two horizontal generatrices through the \(z\)-axis, as expected.

6. Surfaces of revolution

Rotate a curve in a coordinate plane around a coordinate axis lying in that plane. Each point of the curve traces a circle around the axis. The key observation: rotation keeps the coordinate along the axis and keeps the distance from the axis. So in the equation of the curve, keep the axial coordinate, and replace the other coordinate by the distance from the axis.

5° Surfaces of revolution about coordinate axes

Rotating curve	Axis	Surface of revolution
\( F(x, y) = 0,\ z = 0 \)	\(Ox\)	\( F\big(x, \pm\sqrt{y^2 + z^2}\big) = 0 \)
\( F(x, y) = 0,\ z = 0 \)	\(Oy\)	\( F\big(\pm\sqrt{x^2 + z^2}, y\big) = 0 \)
\( F(x, z) = 0,\ y = 0 \)	\(Ox\)	\( F\big(x, \pm\sqrt{y^2 + z^2}\big) = 0 \)
\( F(x, z) = 0,\ y = 0 \)	\(Oz\)	\( F\big(\pm\sqrt{x^2 + y^2}, z\big) = 0 \)
\( F(y, z) = 0,\ x = 0 \)	\(Oy\)	\( F\big(y, \pm\sqrt{x^2 + z^2}\big) = 0 \)
\( F(y, z) = 0,\ x = 0 \)	\(Oz\)	\( F\big(\pm\sqrt{x^2 + y^2}, z\big) = 0 \)

The \(\pm\) matters The book's table writes only \(+\sqrt{\ }\). That is right when the curve is symmetric with respect to the axis (\(F\) is even in the replaced variable), or when only the half of the curve on one side is meant. In general a point at distance \(\rho\) from the axis can come from either side of it, so both signs are needed, and the usual way to clear them is to square. For example, rotating the line \( z = x \) (in the plane \(y = 0\)) about \(Oz\): with \(+\) only, \( z = \sqrt{x^2 + y^2} \) is the upper half-cone; the whole line gives \( z = \pm\sqrt{x^2 + y^2} \), i.e. the double cone \( z^2 = x^2 + y^2 \).

Example 5 · Familiar surfaces of revolution

The parabola \( z = x^2 \) (in \(y = 0\)) about \(Oz\): \( z = x^2 + y^2 \), a paraboloid of revolution (satellite dish).
The ellipse \( \frac{x^2}{a^2} + \frac{z^2}{c^2} = 1 \) about \(Oz\): \( \frac{x^2 + y^2}{a^2} + \frac{z^2}{c^2} = 1 \), a spheroid (the Earth's shape, flattened at the poles when \(c < a\)).
The hyperbola \( \frac{x^2}{a^2} - \frac{z^2}{c^2} = 1 \) about \(Oz\) (the axis not meeting it): \( \frac{x^2 + y^2}{a^2} - \frac{z^2}{c^2} = 1 \), a hyperboloid of one sheet (cooling tower). About \(Ox\) (the axis through its branches): \( \frac{x^2}{a^2} - \frac{y^2 + z^2}{c^2} = 1 \), a hyperboloid of two sheets.

7. Quadric surfaces (canonical forms)

Surfaces given by second-degree equations in \(x, y, z\) are called quadrics, or second-order surfaces. Just as for conics in the plane (see the post on affine transformations), a suitable translation and rotation brings every quadric to a simple canonical form. The non-degenerate ones are:

6° Second-order surfaces (canonical forms)

Surface	Canonical equation	Sections \(z = h\)
1) Ellipsoid	\( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \)	ellipses for \(\|h\| < c\), a point for \(\|h\| = c\)
2) Hyperboloid of one sheet	\( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \)	ellipses for every \(h\), smallest at \(h = 0\)
Hyperboloid of two sheets	\( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1 \)	empty for \(\|h\| < c\), ellipses for \(\|h\| > c\)
3) Elliptic paraboloid	\( \frac{x^2}{p} + \frac{y^2}{q} = 2z \)	ellipses for \(h > 0\)
Hyperbolic paraboloid	\( \frac{x^2}{p} - \frac{y^2}{q} = 2z \)	hyperbolas (two lines for \(h = 0\))
4) Cone of the second order	\( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0 \)	ellipses (a point for \(h = 0\))

(\(a, b, c, p, q > 0\).)

Figure 2. The six non-degenerate quadric surfaces, drawn as wireframes of their horizontal sections and generating curves.

How to recognize and imagine them

Count the signs. All three squares positive and \(= 1\): bounded, an ellipsoid. One minus sign: a hyperboloid; \(= +1\) gives one sheet (connected, like a waist), \(= -1\) gives two sheets (two separate cups). Equal to \(0\): a cone. Only two squares and a linear term: a paraboloid.
Slice it. The best way to picture a quadric is through its sections by planes \(z = h\), \(x = h\), \(y = h\): each section is a conic (an ellipse, hyperbola, parabola or pair of lines). The table lists the horizontal ones.
The saddle. The hyperbolic paraboloid is a saddle (or a Pringles chip): along \(x\) it curves up, along \(y\) down. At the origin it has a stationary point that is not an extremum, exactly the saddle point from the test \(D = AC - B^2 < 0\).
Straight lines on curved surfaces. The hyperboloid of one sheet and the hyperbolic paraboloid are doubly ruled: through each of their points pass two straight lines lying entirely on the surface. That is why cooling towers and some roofs can be built from straight beams.

Example 6 · Lines on the hyperbolic paraboloid For \( z = xy \) (a hyperbolic paraboloid rotated by \(45^\circ\)), the line \( x = 1,\ z = y \) lies entirely on the surface (check: \(xy = y = z\)). In general, \( x = k,\ z = ky \) for each \(k\), and \( y = k,\ z = kx \): two families of straight lines.

Besides these six, the degenerate quadrics are the cylinders over conics (elliptic \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), hyperbolic \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), parabolic \(y^2 = 2px\)) and pairs of planes, as in Section 3.

8. Summary

Surface	Generated by	How to get the equation
Sphere	points at distance \(R\) from \(C\)	\( (x - a)^2 + (y - b)^2 + (z - c)^2 = R^2 \)
Cylinder	line \(\parallel \vec p\) sliding along \((c)\)	eliminate \(M_1\) from line + directrix
Cone	line through \(P\) sliding along \((c)\)	eliminate \(M_1\); vertex at \(O\) gives a homogeneous equation
Conoid	line \(\parallel \Pi\) meeting \(p\) and \((c)\)	\( \varphi(a, b) = 0 \Rightarrow \varphi(y/x, z) = 0 \)
Revolution	curve rotating about an axis	other coordinate \(\to \pm\)distance from the axis
Quadrics	second-degree equations	canonical forms; picture through sections

9. Practice problems

Try each one before opening the solution. More solved problems from this topic are in Geometry.

Problem 1: Find the centre and radius of \( x^2 + y^2 + z^2 + 6z = 0 \).

\( x^2 + y^2 + (z + 3)^2 = 9 \): centre \((0, 0, -3)\), radius \(3\). (It passes through the origin.)

Problem 2: What surface is \( y^2 = 4x \) in space?

The variable \(z\) is missing: a parabolic cylinder with generatrices parallel to \(Oz\), over the parabola \(y^2 = 4x\) in the plane \(z = 0\).

Problem 3: Find the cylinder with directrix \( x^2 + y^2 = 4,\ z = 0 \) and generatrices parallel to \( \{0, 1, 2\} \).

\( \frac{x - x_1}{0} = \frac{y - y_1}{1} = \frac{z}{2} \): \( x_1 = x \), \( y_1 = y - \frac z2 \). Equation: \( x^2 + \big(y - \frac z2\big)^2 = 4 \).

Problem 4: Find the cone with vertex \(O\) and directrix \( \frac{x^2}{4} + \frac{y^2}{9} = 1,\ z = 2 \).

\( x_1 = \frac{2x}{z} \), \( y_1 = \frac{2y}{z} \): \( \frac{x^2}{z^2} + \frac{4y^2}{9z^2} = 1 \), i.e. \( 9x^2 + 4y^2 - 9z^2 = 0 \). A second-order cone, as expected.

Problem 5: Rotate the circle \( (x - 3)^2 + z^2 = 1 \) (in \(y = 0\)) about \(Oz\).

Replace \(x\) by \(\pm\sqrt{x^2 + y^2}\): \( \big(\sqrt{x^2 + y^2} - 3\big)^2 + z^2 = 1 \) (the sign \(-\) gives nothing new because the circle lies at \(x > 0\)). This is a torus. Without roots: \( (x^2 + y^2 + z^2 + 8)^2 = 36(x^2 + y^2) \).

Problem 6: Identify \( x^2 - y^2 - z^2 = 1 \).

Two minus signs with \(= 1\) means one minus sign with \(= -1\) after multiplying by \(-1\): \( -x^2 + y^2 + z^2 = -1 \). A hyperboloid of two sheets with axis \(Ox\) (sections \(x = h\) are empty for \(|h| < 1\)). It is a surface of revolution: rotate \(x^2 - y^2 = 1\) about \(Ox\).

Final thoughts

Most surfaces in this chapter are generated by motion: a line sliding along a curve (cylinder), a line pivoting at a point (cone), a horizontal line climbing a pole and a railing (conoid), a curve spinning around an axis (surface of revolution). The method for finding their equations is always the same: describe one position of the moving object, use the condition that it touches the guiding curve, and eliminate the coordinates of the point of contact. The quadrics then form a small, complete catalogue of second-degree surfaces, best understood by slicing them into conics.

Related posts: the plane, the line in space, affine transformations and conics, and surfaces of revolution (their areas).