Computes the (signed) Area Under the Curve (AUC) of a path, defined by vectors of x and y coordinates, as compared to an ideal line passing through the start and end points.

## Arguments

- x_vector
x-coordinates of the executed path.

- y_vector
y-coordinates of the executed path.

- x_start
x-coordinate of the start point of the ideal line. Defaults to the first value in

`x_vector`

.- y_start
y-coordinate of the start point of the ideal line. Defaults to the first value in

`y_vector`

.- x_end
x-coordinate of the end point of the ideal line. Defaults to the last value in

`x_vector`

.- y_end
y-coordinate of the end point of the ideal line. Defaults to the last value in

`y_vector`

.- geometric
Whether the sign of areas that stem from a movement in the reverse direction of the ideal line should be reversed. Defaults to

`FALSE`

, indicating an time-based instead of geometric interpretation. Only impacts the AUC if the trajectory is not monotonically increasing relative to the ideal line.

## Details

The ideal line is a line, not a line segment, i.e., it has infinite length. The supplied vectors are assumed to be ordered by time. Counterclockwise deviations from the ideal line are considered positive, clockwise deviations as negative for the computation of the AUC. Thus, negative AUCs are possible.

## References

Wirth, R., Foerster, A., Kunde, W., & Pfister, R. (2020). Design choices: Empirical recommendations for designing two-dimensional finger tracking experiments. Behavior Research Methods, 52, 2394 - 2416. doi:10.3758/s13428-020-01409-0

## Examples

```
x_vals <- c(0, 0, 0, 1, 2)
y_vals <- c(0, 1, 2, 2, 2)
plot(x_vals, y_vals, type = "l")
lines(c(0, 2), c(0, 2), lty = "dashed", lwd = 2) # ideal
auc(x_vals, y_vals) # counterclockwise deviation: positive
#> [1] 2
x_vals <- c(0, 1, 2, 2, 2)
y_vals <- c(0, 0, 0, 1, 2)
auc(x_vals, y_vals) # clockwise deviation: negative
#> [1] -2
plot(x_vals, y_vals, type = "l")
lines(c(0, 2), c(0, 2), lty = "dashed", lwd = 2) # ideal
x_vals <- -x_vals
auc(x_vals, y_vals) # now it is counterclockwise again
#> [1] 2
x_vals <- c(0, 0, 1, 2, 2)
y_vals <- c(0, 1, 1, 1, 2)
plot(x_vals, y_vals, type = "l")
lines(c(0, 2), c(0, 2), lty = "dashed", lwd = 2) # ideal
auc(x_vals, y_vals) # might create small rounding errors; this should be 0
#> [1] 3.330669e-16
all.equal(0, auc(x_vals, y_vals)) # indeed interpreted by R as basically 0
#> [1] TRUE
x_vals <- c(0, 1, 2, 1)
y_vals <- c(0, 1, 1, 0)
plot(x_vals, y_vals, type = "l")
lines(c(0, 1), c(0, 0), lty = "dashed", lwd = 2) # ideal
auc(x_vals, y_vals)
#> [1] 2
auc(x_vals, y_vals, geometric = TRUE) # note the difference
#> [1] 1
```