MCM-ICM-ski-trail

Data available from google elevation API

Need a poltly acount for fancy visulization.

Step 1: Use Google Elevation API to download data.

Step 2: Transform raw data into 2D array map format.

Step 3: Ski Trace Searching Algorithm

def Search(Point a, step r, slope s):
    find all the point(in 36 different direction) on the cycle centered at a, radius = r.
    claculate the slope
    If (all slope is negative):
        return
    Else:
        find the 3 point with slope closet to s
        a.record(b,c,d)
        search(b),search(c), search(d)

Slope is $tan(\theta)$, which is the ratio between horizontal and vertical distance.

Optimization:

• Keep tracking the variance of slopes on each ski track, terminate the search in advance for those with high variance.

• recurrence with memorization to avoid repited computation.

Step 4: Random Start Searching, Minimize the Cost function.

Among the paths we randomly searched, our goal is to find those satisfy two objective:

• The trail slope is close to the given area

The hinge loss is applied, there's a linear penalty for the slopes exceeding:

$$ f(i)= \begin{cases} k-S_{i} & \text{ if } S_{i}<k \\ \infty & \text{ if } S_{i}>k' \end{cases} $$

$$ L_1 = \sum_{i=1}^{N}f(i)$$

• The terrian roughness is as small as possible

We use the standard deviation of slopes on the path:

$$L_2 = \sigma_{a} ={\sqrt {{\frac {1}{N}}\sum {i=1}^{N}(S{i}-\bar{S})^{2}}},{\rm {\ \ where\ \ }}\bar{S} ={\frac {1}{N}}\sum {i=1}^{N}S{i}$$

In order to get the overall loss function, we normalize each of above objective and add them together.

4.1,2,3 Find Optimal paths for easy,intermidiate,hard trail

randomly choose 5000 starting point and find the coresspoing paths.

Find 10 best path with the smallest cost