Angular Wedges And Spherical Sectors

Two Different Kinds of Angular Restriction

A region can be restricted by an angle in two fundamentally different ways. An azimuthal restriction selects directions around the \(z\)-axis, like choosing a slice of a pie viewed from above. A polar restriction selects directions measured down from the positive \(z\)-axis, like selecting a band of opening directions around that axis.

These restrictions use different spherical coordinates. In the convention used here, \(\theta\) is the azimuthal angle in the \(xy\)-plane, while \(\phi\) is the polar angle measured from the positive \(z\)-axis. Keeping these roles distinct makes it easier to describe wedges and conical sectors correctly.

Coordinate Conventions and Definitions

Cylindrical coordinates are \((r,\theta,z)\), where \(r\geq 0\) is the distance from the \(z\)-axis and \(\theta\) gives the direction around that axis. Spherical coordinates are \((\rho,\theta,\phi)\), with \(\rho\geq 0\) the distance from the origin. Their Cartesian formulas are

x=\rho\sin\phi\cos\theta,\qquad y=\rho\sin\phi\sin\theta,\qquad z=\rho\cos\phi.

The angle ranges are \(0\leq\theta<2\pi\) and \(0\leq\phi\leq\pi\). The corresponding cylindrical and spherical coordinates satisfy

r=\rho\sin\phi,\qquad z=\rho\cos\phi.

Definition: An azimuthal wedge is a region whose directions around the \(z\)-axis are restricted by bounds \(\alpha\leq\theta\leq\beta\). A conical sector is a region whose directions from the origin are restricted by polar-angle bounds \(\gamma\leq\phi\leq\delta\). Bounds on \(r\), \(\rho\), or \(z\) may further limit the region, but do not change which angle is being restricted.

For a wedge, the angle \(\theta\) is measured in the \(xy\)-plane from the positive \(x\)-axis. The wedge extends in the \(z\)-direction unless an additional condition limits it. For a conical sector, \(\phi\) measures inclination from the positive \(z\)-axis: \(\phi=0\) points straight upward, \(\phi=\pi/2\) lies in the \(xy\)-plane, and \(\phi=\pi\) points straight downward.

At the \(z\)-axis, \(r=0\), so \(\theta\) is not uniquely determined. At the origin, \(\rho=0\), and neither angle determines a unique direction. When describing a wedge or sector that reaches these locations, it is usual to include the axis or origin when it is a limiting point of the stated region. The angular bounds describe directions away from those points.

Describing an Azimuthal Wedge

A cylindrical wedge is described directly by bounding \(\theta\). For example, \(0\leq\theta\leq\pi/4\), \(0\leq r\leq 2\), with \(z\) unrestricted, describes a vertical wedge inside a cylinder. In spherical coordinates, \(\theta\) is unchanged: the same angular restriction applies, while \(r\leq2\) becomes \(\rho\sin\phi\leq2\).

This is the key fact for azimuthal restrictions: the cylindrical and spherical descriptions use the same \(\theta\). The conversion changes the radial distance from the axis, \(r\), into \(\rho\sin\phi\), but it does not turn the wedge angle into a polar-angle bound.

Worked Example: Convert a Vertical Wedge to Spherical Coordinates

Describe the region \(0\leq\theta\leq\pi/4\), \(0\leq r\leq2\), with no restriction on \(z\), in spherical coordinates.

The azimuthal condition remains \(0\leq\theta\leq\pi/4\). Since \(r=\rho\sin\phi\), the radial bound becomes \(0\leq\rho\sin\phi\leq2\). Because \(\rho\geq0\) and \(0\leq\phi\leq\pi\), \(\sin\phi\geq0\), so the lower bound is automatic. There is no restriction on \(z\), and therefore no additional bound on \(\phi\) or \(\rho\).

0\leq\theta\leq\frac{\pi}{4},\qquad \rho\sin\phi\leq2,\qquad \rho\geq0,\quad 0\leq\phi\leq\pi.

To check the conversion, take \(\rho=2\), \(\phi=\pi/4\), and \(\theta=\pi/4\). Then \(r=2\sin(\pi/4)=\sqrt{2}\), which satisfies \(0\leq r\leq2\), and \(\theta\) satisfies its original bound. The corresponding Cartesian point is \(x=1\), \(y=1\), \(z=\sqrt{2}\), since \(2(\sqrt{2}/2)(\sqrt{2}/2)=1\) for each of \(x\) and \(y\), and \(2(\sqrt{2}/2)=\sqrt{2}\) for \(z\). It lies in the stated wedge.

A wedge can also be recognized from Cartesian inequalities in the horizontal plane. For instance, \(0\leq\theta\leq\pi/4\) describes the sector between the positive \(x\)-axis and the line \(y=x\), on the side where \(y\geq0\). The angle restriction selects directions; a separate radial bound controls how far from the axis the region extends.

Describing a Conical Sector

A conical sector is specified by bounds on \(\phi\), not on \(\theta\). For \(0<\gamma<\delta<\pi\), the bounds \(\gamma\leq\phi\leq\delta\) select the directions between two cones with vertex at the origin. If \(\theta\) is unrestricted, the region sweeps all the way around the \(z\)-axis. Bounds on \(\rho\), such as \(0\leq\rho\leq R\), can limit the sector to a finite distance from the origin.

When \(0<\gamma<\delta<\pi\), the same polar-angle restriction can also be written in cylindrical coordinates. For \(r>0\), the relations give \(z/r=\cot\phi\). Since \(\cot\phi\) decreases as \(\phi\) increases on \((0,\pi)\), the order of the endpoint bounds reverses.

Result: For \(r>0\) and \(0<\gamma<\delta<\pi\), \(\gamma\leq\phi\leq\delta\) is equivalent to \(r\cot\delta\leq z\leq r\cot\gamma\). If the sector also has \(0\leq\rho\leq R\), that radial bound becomes \(r^2+z^2\leq R^2\).

The cylindrical inequalities describe the region between the two bounding cones. If the angular interval includes a pole, \(\phi=0\) or \(\phi=\pi\), the corresponding direction lies on the \(z\)-axis, where \(r=0\); handle that axis as a limiting direction rather than relying on \(z/r\).

Worked Example: Convert an Upper Conical Sector to Cylindrical Coordinates

Convert the spherical sector \(0\leq\rho\leq4\), \(\pi/6\leq\phi\leq\pi/3\), with \(\theta\) unrestricted, into cylindrical coordinates.

First convert the spherical radial bound. Since \(\rho^2=r^2+z^2\), the condition \(0\leq\rho\leq4\) is equivalent to \(r^2+z^2\leq16\), with \(r\geq0\). Next use the conical-sector result. The lower cylindrical bound is \(r\cot(\pi/3)=r/\sqrt{3}\), because \(\cot(\pi/3)=1/\sqrt{3}\). The upper bound is \(r\cot(\pi/6)=\sqrt{3}r\), because \(\cot(\pi/6)=\sqrt{3}\). Thus the region is described by

r\geq0,\qquad r^2+z^2\leq16,\qquad \frac{r}{\sqrt{3}}\leq z\leq\sqrt{3}\,r,

with every azimuthal direction allowed. To check the conversion, choose \(\rho=2\), \(\phi=\pi/4\), and \(\theta=0\). Then \(r=2\sin(\pi/4)=\sqrt{2}\) and \(z=2\cos(\pi/4)=\sqrt{2}\). The spherical bounds hold because \(0\leq2\leq4\) and \(\pi/6\leq\pi/4\leq\pi/3\). In cylindrical coordinates, \(r^2+z^2=2+2=4\leq16\), while \(r/\sqrt{3}=\sqrt{6}/3\leq\sqrt{2}=z\leq\sqrt{6}=\sqrt{3}r\). Substituting \(r=z=\sqrt{2}\) into \(z/r\) gives \(1=\cot(\pi/4)\), confirming the angle.

Comparing Azimuthal and Polar Bounds

The two types of angular restriction behave differently as a region extends around the \(z\)-axis. A fixed \(\theta\) selects one vertical half-plane through that axis. Varying \(\phi\) within that half-plane changes the inclination of a ray, but does not rotate the half-plane. Conversely, fixing \(\phi\) and allowing \(\theta\) to vary sweeps a cone all the way around the axis.

Restriction	Angle meaning	Typical geometric effect
\(\alpha\leq\theta\leq\beta\)	Direction around the \(z\)-axis in the \(xy\)-plane	A vertical wedge or angular slice
\(\gamma\leq\phi\leq\delta\)	Inclination from the positive \(z\)-axis	A conical band of directions
\(\theta\) unrestricted, \(\gamma\leq\phi\leq\delta\)	Polar inclination only	A sector swept around the full axis

Worked Example: Recognize a Polar Restriction in the Lower Half-Space

Convert the sector \(0\leq\rho\leq3\), \(2\pi/3\leq\phi\leq5\pi/6\), with \(\theta\) unrestricted, to cylindrical coordinates.

The radial bound becomes \(r^2+z^2\leq9\). For the angular bounds, \(\cot(5\pi/6)=-\sqrt{3}\) and \(\cot(2\pi/3)=-1/\sqrt{3}\). Therefore the cylindrical inequalities are \(-\sqrt{3}r\leq z\leq-r/\sqrt{3}\). In particular, \(z\) is negative when \(r>0\), as expected for angles in this range, which lies below the \(xy\)-plane.

r\geq0,\qquad r^2+z^2\leq9,\qquad -\sqrt{3}\,r\leq z\leq-\frac{r}{\sqrt{3}},

To verify, take \(\rho=2\), \(\phi=3\pi/4\), and \(\theta=0\). Then \(r=2\sin(3\pi/4)=\sqrt{2}\) and \(z=2\cos(3\pi/4)=-\sqrt{2}\). The spherical conditions hold because \(0\leq2\leq3\) and \(2\pi/3\leq3\pi/4\leq5\pi/6\). In cylindrical form, \(r^2+z^2=2+2=4\leq9\). Also, \(-\sqrt{3}r=-\sqrt{6}\leq-\sqrt{2}=z\), and \(-\sqrt{2}\leq-\sqrt{2/3}=-r/\sqrt{3}\). Finally, \(z/r=-1=\cot(3\pi/4)\), which recovers the chosen polar angle.

Why the Cylindrical Inequalities Work

The conversion from polar-angle bounds to bounds on \(z/r\) follows directly from the coordinate definitions. For \(r>0\), the cylindrical-spherical relation gives \(r=\rho\sin\phi\) and \(z=\rho\cos\phi\). Because \(0<\phi<\pi\) implies \(\sin\phi>0\), division is valid, and

\frac{z}{r} = \frac{\rho\cos\phi}{\rho\sin\phi} = \cot\phi.

On \((0,\pi)\), the function \(\cot\phi\) is strictly decreasing: its derivative is \(-\csc^2\phi\), which is negative wherever the function is defined. Thus if \(\gamma\leq\phi\leq\delta\), applying \(\cot\) reverses the inequalities, giving \(\cot\delta\leq\cot\phi\leq\cot\gamma\). Multiplying by \(r>0\) yields \(r\cot\delta\leq z\leq r\cot\gamma\). Each step preserves equivalence, so the cylindrical inequalities describe exactly the same angular range for points with \(r>0\).

Common Mistake: Interchanging \(\theta\) and \(\phi\)

A frequent error is to treat a polar restriction as if it were an azimuthal one. For example, the point \((x,y,z)=(0,1,\sqrt{3})\) has \(r=1\), \(\rho=2\), \(\theta=\pi/2\), and \(\phi=\pi/6\). It satisfies a sector restriction \(0\leq\phi\leq\pi/4\). But it does not satisfy \(0\leq\theta\leq\pi/4\), because \(\pi/2>\pi/4\).

The incorrect calculation would replace the intended polar condition by a bound on \(\theta\), then wrongly exclude this point. The correction is to identify what the angle measures before applying a bound: \(\phi\) controls inclination from the positive \(z\)-axis, while \(\theta\) controls direction around that axis. A region with a \(\phi\)-restriction and unrestricted \(\theta\) goes around the axis; a \(\theta\)-restriction selects only part of that rotation.

Using Angular Descriptions Reliably

When translating a region, keep the angular and radial conditions separate. First record whether the restriction is on \(\theta\) or \(\phi\). Preserve \(\theta\) when moving between cylindrical and spherical coordinates. For a \(\phi\)-restriction with \(0<\gamma<\delta<\pi\), use the reversed cotangent bounds in cylindrical coordinates. Then convert any distance bound using \(r=\rho\sin\phi\) or \(\rho^2=r^2+z^2\), as appropriate.

These distinctions matter because a small change in the angle named can change the region completely: a vertical wedge and a sector between cones are not interchangeable descriptions. Checking one representative point, as in the examples, is a useful way to catch a mistaken angular interpretation.

Check Your Understanding

For each question, identify which angle is restricted and use the coordinate relationships to justify your answer.

What geometric directions are selected by \(0\leq\theta\leq\pi/3\) when \(z\) is unrestricted?
Write the spherical description of \(0\leq r\leq5\), \(\pi/4\leq\theta\leq\pi/2\), with \(z\) unrestricted.
Convert \(\pi/6\leq\phi\leq\pi/3\), for \(r>0\), into cylindrical inequalities involving \(z\) and \(r\).
Why do the cotangent bounds reverse the order of the polar-angle endpoints?
A region has unrestricted \(\theta\) and \(2\pi/3\leq\phi\leq5\pi/6\). Is it an azimuthal wedge or a conical sector, and in which half-space does it lie away from the axis?