4.  Consider spherical surface $S$ which radius is $1$, central point $(0,0,1)$ in $xyz$ space.  If point $Q$ move to points on S expect $(0,0,2)$.  Let $R$ be an intersection of plane $z=0$ and line $l$ pass point $Q$ and point $P (1,0,2)$.  Find the range of moving $R$, then illustrate it.
1. **Define the spherical surface \( S \):**
   The spherical surface \( S \) has a radius of 1 and is centered at the point \((0,0,1)\). Therefore, any point \( Q \) on the surface \( S \) satisfies the equation:
   \[
   (x-0)^2 + (y-0)^2 + (z-1)^2 = 1
   \]
   Simplifying, we get:
   \[
   x^2 + y^2 + (z-1)^2 = 1
   \]

2. **Parameterize the point \( Q \):**
   Let \( Q \) be \((a, b, c)\) such that:
   \[
   a^2 + b^2 + (c-1)^2 = 1 \quad \text{and} \quad 0 \leq c < 2
   \]

3. **Find the equation of the line \( l \) passing through points \( Q \) and \( P(1,0,2) \):**
   The parametric form of the line \( l \) can be written as:
   \[
   \mathbf{r}(t) = (1 + t(a-1), t b, 2 + t(c-2)), \quad t \in \mathbb{R}
   \]

4. **Find the intersection \( R \) of the line \( l \) with the plane \( z = 0 \):**
   Set \( z = 0 \) in the parametric equation:
   \[
   2 + t(c-2) = 0
   \]
   Solving for \( t \):
   \[
   t = \frac{-2}{c-2} = \frac{2}{2-c}
   \]

5. **Substitute \( t \) back into the parametric equations for \( x \) and \( y \):**
   \[
   x = 1 + \frac{2(a-1)}{2-c} = \frac{2a - c}{2-c}
   \]
   \[
   y = \frac{2b}{2-c}
   \]

6. **Determine the locus of \( R \):**
   The coordinates of \( R \) are:
   \[
   R \left( \frac{2a - c}{2-c}, \frac{2b}{2-c} \right)
   \]

7. **Express the locus in terms of \( p \):**
   Let \( p = \frac{c}{2-c} \). Then:
   \[
   x = \frac{2a - c}{2-c} = 2a - c \cdot \frac{1}{2-c} = 2a - p
   \]
   \[
   y = \frac{2b}{2-c} = 2b \cdot \frac{1}{2-c} = 2b
   \]

8. **Find the envelope of \( R \):**
   The envelope of \( R \) is given by the equation:
   \[
   (x + p)^2 + y^2 = 4p
   \]
   Substituting \( p = \frac{c}{2-c} \):
   \[
   (x + \frac{c}{2-c})^2 + y^2 = 4 \cdot \frac{c}{2-c}
   \]

9. **Simplify the envelope equation:**
   \[
   y^2 + 4x - 4 = 0
   \]

10. **Determine the range of \( R \):**
    The range of \( R \) is given by:
    \[
    y^2 + 4x - 4 \leq 0
    \]

The final answer is \( \boxed{ y^2 + 4x - 4 \leq 0 } \)