Assume we have GPS coordinates, simplified as coordinate triples $(7,5,19),$ each Write functions as follows

distanceVertical$($gpstrack$)$ function

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Function distance Horizontal $($gps track$)$ Statistics $1$ of $1$ tests passing

Should calculate and return the horizontal distance traveled $($in meters$).$

Function Plan $($GPSTrack$)$

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The function should return the predicted walking time in minutes; it does not need to be rounded $($see above for calculation basis$).$

function calcspeed $($gpstrack$)$

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Function breakTime$($gpstrack$,$ maxdelta$)$

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Function numberBreaks$($gpstrack$,$ maxdelta, minseconds$)$

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This function should count all pauses that are at least minseconds long and return this number.

Function smooth$($gpstrack$,$ weights$)$

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Should return a new gps track that is calculated from what was passed by smoothing with the weights passed.

weights do not have to be passed and should be $0\mathrm{.}33$ by default.

Only filters with length $3$ are used $($i$.$e$.$ current point and one before and after$).$

Function speedDifference$($gpstrack$,$ weightsA, weightsB$)$

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AThe weights should have default values, namely unsmoothed $($i$.$e$.$ weights $,1,0)$ for weightsA and averaging $(\mathrm{.}33,...)$ for weightsB.specifying the x$,$ y$,$ and z coordinates of a point. To simplify, we assume the earth is flat, x$,$y indicate the location and z the height. The coordinates are absolute and are given in meters.

The GPS path $[(0,0,0),(5,0,0,),(10,0,0)]$ describes that the path led $10$m to the east.

Furthermore, the following assumptions apply:

A point is recorded every $2$ seconds.

Points ignore the curvature of the earth.

Point entries are in meters.

A point always has three coordinates $($x$,$y$,$z$).$

A GPS track can contain zero, one or more points.

A GPS track only contains valid points $($so no check for $"$is that a number" or similar necessary$).$

Route calculation

When calculating horizontal movement, the Z coordinate is ignored; when calculating spatial movement, it is included. The vertical movement uses only the Z coordinate.

To calculate horizontal motion and spatial motion, use the direct path $($Pythagoras$).$ For this you need the square root. It is available in the math module $($import math$)$:

math.sqrt$($x$)$ : Returns the square root of x$.$

Break

A break is assumed when two points do not differ by more than delta in the spatial diagonal $($i$.$e$.$ there is only a slight measurement wobble as movement$).$Delta$$ only applies between two neighboring points. So if points A$-$B$-$C represent a $4$ second pause $($remember$,$ one point is taken every two seconds$),$ then A differs from B by no more than delta and B differs from C as well. A and C can be more than delta apart!

Calculation of walking time for hikers

Use the following information to calculate walking times:

The following assumptions are made in DIN $33466$: A hiker covers in one hour:

$300$ meters in ascent.

$500$ meters in descent.

$4$ kilometers horizontal distance.

The actual walking time for a route can be calculated by halving the smaller of the times calculated for the horizontal and vertical distance and adding it to the larger one.

Example: Difference in altitude: $900$m $900$m$/300$m $->3$ h Horizontal distance: $8$ km $8$ km$/4$ km $->2$ h How it's calculated: $2\mathrm{.}0$ h x $0\mathrm{.}5+3\mathrm{.}0$ h $=4\mathrm{.}0$ h Result: The walking time is therefore $4$ hours.

Example $2$: Like example $1$ but with a return path: $900$m ascent $(3$h$)$ and descent $($~$1$h $45$m $(1\mathrm{.}8$h$))$: $4$h $45$m $16$km Horizontal: $4$h Result: $4$h $\backslash$times $0\mathrm{.}5+4$h $45$m $=6$h $45$m $($or $6\mathrm{.}8$h$)$

Filter for smoothing

Measurements of GPS coordinates are naturally subject to measurement fluctuations. This fluctuation is included in the calculation as an extra distance. If we want to avoid this, we can smooth the GPS track by averaging three neighboring values $($moving average of current, previous and next point$).$ At the beginning and end of the track we are one point short, so these points are simply not filtered.

A slightly better version of this smoothing is to use weights, i$.$e$.$ the points are not all worth

$1$

$3$

$3$

$1$

$$

but for example g $=[0\mathrm{.}2,0\mathrm{.}6,0\mathrm{.}2].$ So the current point has the strongest weight and contributes $60\%(0\mathrm{.}6),$ the points before and after only contribute $20\%(0\mathrm{.}2).$

Example:

gpstrack $=[[1,1,1],[4,2,1],[10,0,2]]$

g $=[0\mathrm{.}2,0\mathrm{.}6,0\mathrm{.}2]$

gpstracksmooth $=[$

$[1,1,1],$ # unfiltered because there is no previous point

$[0\mathrm{.}2*1+0\mathrm{.}6*4+0\mathrm{.}2*10,$ # x point $2$

$0\mathrm{.}2*1+0\mathrm{.}6*2+0\mathrm{.}2*0,$ # y point $2$

$0\mathrm{.}2*1+0\mathrm{.}6*1+0\mathrm{.}2*2],$# z point $2$

$[10,0,2]$ # unfiltered because there is no next point

$]$

Please note that the x$,$ y$,$ and z coordinates are processed and weighted independently of each other! So the new x coordinate is just a combination of x coordinates, etc.