For both rendering and collision detection with rocks and the hiker, the spline-based representation of the terrain is impractical.
Therefore, the splines are approximated differently for each purpose:
To make the terrain visually appealing, the ground $H(t)$ is approximated by a polyline $T_{rend}$ with a relatively high resolution $res_{rend}$:
$$ T_{rend} = \left(H(0), H(res_{rend}), H(2\cdot res_{rend}), \dots, H\left(\left\lfloor{\frac{t_n}{res_{rend}} - 1}\right\rfloor\cdot res_{rend}\right), H(t_n)\right)$$
To improve performance, the ground is approximated by a polyline $T_{i,cd}$ for each spline-piece $H_i(t)$ using a coarser resolution $res_{cd}$. Combining all of these piecewise approximations into one results in the representation $T_{cd}$ for collision detection:
$$\forall i\in{0,1, \dots, n-1}: T_{i, cd} = \left(H(t_i), H(t_i + res_{cd}), H(t_i + 2\cdot res_{cd}), \dots,H\left(t_i + \left\lfloor{\frac{t_{i+1} - t_i}{res_{cd}} - 1}\right\rfloor\cdot res_{cd}\right), H(t_{i+1})\right)$$
This type of segmented representation is particulatrily useful for an efficient preselection of relevant terrain sections in the collision detection.