如何求过定点且与已知3D直线成定角的直线簇方程

Let's break this down step by step to find the set of all lines that meet your criteria. As you noted, these lines are exactly the generators of a cone—with the given line as its axis and the fixed point $(x_1,y_1,z_1)$ as its vertex—having a semi-vertical angle of $ heta$.

Step 1: Use the Angle Between Direction Vectors #

First, remember that the angle between two lines is determined by the angle between their direction vectors. Let's define:

• The given line has direction vector $\boldsymbol{v} = (a_1, b_1, c_1)$ (its direction ratios)
• Any unknown line passing through $(x_1,y_1,z_1)$ will have some direction vector $\boldsymbol{w} = (l, m, n)$ (where $l, m, n$ can't all be zero)

The angle $ heta$ between the two lines gives us this key relation from the dot product formula:
$$\cos heta = \frac{|\boldsymbol{v} \cdot \boldsymbol{w}|}{|\boldsymbol{v}| \cdot |\boldsymbol{w}|}$$

Step 2: Derive the Condition for Valid Direction Ratios #

To eliminate the absolute value and simplify, square both sides of the equation above:
$$\cos^2 heta = \frac{(\boldsymbol{v} \cdot \boldsymbol{w})^{2}{|\boldsymbol{v}|}2 \cdot |\boldsymbol{w}|^2}$$

Multiply both sides by $|\boldsymbol{v}|^2 \cdot |\boldsymbol{w}|^2$ to get:
$$|\boldsymbol{v}|^2 \cdot |\boldsymbol{w}|^2 \cdot \cos^2 heta = (\boldsymbol{v} \cdot \boldsymbol{w})^2$$

Substitute in the components of the vectors:
$$(a_1^2 + b_1^2 + c_1^2)(l2 + m^2 + n^2)\cos2 heta = (a_1 l + b_1 m + c_1 n)^2$$

We can expand and simplify this using $\sin^2 heta = 1 - \cos^2 heta$ to get a cleaner quadratic equation for $(l,m,n)$:
$$(b_1^2 + c_1^2)\cos2 heta \cdot l^2 + (a_1^2 + c_1^2)\cos2 heta \cdot m^2 + (a_1^2 + b_1^2)\cos2 heta \cdot n^2 - 2a_1b_1\sin^2 heta \cdot lm - 2a_1c_1\sin^2 heta \cdot ln - 2b_1c_1\sin^2 heta \cdot mn = 0$$

This equation defines all valid direction ratios $(l,m,n)$ for the lines we're looking for.

Step 3: Parametric Form of All Satisfying Lines #

Every line that meets the problem's requirements can be written in parametric form as:
$$\boldsymbol{r} = (x_1, y_1, z_1) + t(l, m, n)$$
where:

• $t \in \mathbb{R}$ is a parameter (any real number)
• $(l, m, n)$ is any non-zero triplet of real numbers that satisfies the quadratic equation derived in Step 2

In short, this set of lines is exactly the generators of the cone described: vertex at $(x_1,y_1,z_1)$, axis along the given line, semi-vertical angle $ heta$.

内容的提问来源于stack exchange，提问作者Shaleen Jain