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3D Computational Geometry Basics

Many concepts in three-dimensional geometry are analogous to those in two-dimensional geometry, and we can use the same methods for solving 3D geometric problems as we do for 2D ones.

Basic Concepts

Concepts such as points, vectors, and lines are similar to those in 2D geometry and will not be elaborated here.

Plane

A plane can be represented by a point \(P_0(x_0,y_0,z_0)\) on the plane and its normal vector (i.e., a vector perpendicular to the plane) \(\boldsymbol{n}\).

Since \(\boldsymbol{n}\) is perpendicular to the plane, it is perpendicular to all lines within the plane. In other words, let \(\boldsymbol{n}=(A,B,C)\). Then any point \(P(x,y,z)\) on the plane satisfies \(\boldsymbol{n} \cdot \overrightarrow{PP_0} = 0\).

By the definition of the dot product, the above equation is equivalent to:

\[ A(x-x_0)+B(y-y_0)+C(z-z_0)=0 \]

Rearranging gives:

\[ Ax+By+Cz-(Ax_0+By_0+Cz_0)=0 \]

Let \(D=-(Ax_0+By_0+Cz_0)\). Then the equation becomes \(Ax+By+Cz+D=0\). This form is called the general form of a plane.

Basic Operations

Angles Between Lines and Planes

Using knowledge of spatial vectors, the angles between lines and planes in space can be quickly determined.

For two skew lines \(a\) and \(b\), take a point \(P\) in space and draw \(a' \parallel a\) and \(b' \parallel b'\). The acute or right angle between \(a'\) and \(b'\) is called the angle between the two skew lines \(a\) and \(b\).

For a line \(a\) and a plane \(\alpha\), if \(a\) intersects \(\alpha\) at \(A\), take a point \(P\) on \(a\) and drop a perpendicular from \(P\) to \(\alpha\), meeting \(\alpha\) at \(O\). Then the complement of the angle between \(a\) and \(PO\) is called the angle between the line and the plane. In particular, if \(a \parallel \alpha\) or \(a \subset \alpha\), the angle between them is \(0^\circ\).

For two planes \(\alpha\) and \(\beta\), their angle is defined as the angle between two lines \(a\) and \(b\) (with \(a \subset \alpha\), \(b \subset \beta\)) that are both perpendicular to the intersection line \(l\) of the two planes.

Definition of Angle Between Two Lines and Necessary and Sufficient Conditions

  • The angle between the direction vectors of two lines is called the angle between the lines.

With this proposition, we can derive the following: Given two lines \(l_1, l_2\) with direction vectors \(s_1 (m_1, n_1, p_1)\) and \(s_2 (m_2, n_2, p_2)\), let \(\varphi\) be the angle between the lines. Then we have \(\cos \varphi = \dfrac{\left | m_1m_2+n_1n_2+p_1p_2 \right |}{\sqrt{m_1^2+n_1^2+p_1^2}\sqrt{m_2^2+n_2^2+p_2^2}}\).

  • \(l_1 \perp l_2 \iff m_1m_2 + n_1n_2 + p_1p_2 = 0\)

  • \(l_1 \parallel l_2 \iff \dfrac{m_1}{m_2} = \dfrac{n_1}{n_2} = \dfrac{p_1}{p_2}\).

Angle Between a 3D Vector and a Plane

When a line is not perpendicular to a plane, the angle \(\varphi\) (\(\varphi \in [0, \frac{\pi}{2}]\)) between the line and its projection onto the plane is called the angle between the line and the plane.

Let the line's direction vector be \(s(m, n, p)\) and the plane's normal vector be \(f(a, b, c)\). Then the following propositions hold:

  • Sine of the angle: \(\sin\varphi = \dfrac{\left | am + bn + cp \right |}{\sqrt{a^2+b^2+c^2}\sqrt{m^2+n^2+p^2}}\)

  • The line is parallel to the plane \(\iff am+bn+cp = 0\)

  • The line is perpendicular to the plane \(\iff \dfrac{a}{m} = \dfrac{b}{n} = \dfrac{c}{p}\)

Distance from a Point to a Plane

Intersection of a Line and a Plane

Simply solve the system of equations formed by the line equation and the plane equation.

Solid Geometry Theorems

Theorem of Three Sines

Let the dihedral angle \(M－AB－N\) have measure \(\alpha\). On plane \(M\), there is a ray \(AC\) that forms an angle \(\beta\) with the edge \(AB\) and an angle \(\gamma\) with plane \(N\). Then \(\sin\gamma = \sin\alpha\cdot\sin\beta\).

Theorem of Three Cosines

Let \(O\) be a point on a plane. For a point \(B\) outside the plane, the projection of line \(BO\) onto the plane is \(AO\), and \(OC\) is a line in the plane. Then the cosines of angles \(\angle COB\), \(\angle AOC\), and \(\angle AOB\) satisfy: \(\cos\angle BOC=\cos\angle AOB\cdot\cos\angle AOC\) (where \(\angle AOC\) and \(\angle AOB\) must be acute).

References

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