MATLAB Mandible Tracing & Calculations Program

MATLAB Mandible Tracing & Calculations Program Documentation by Antonis Vakis Supplemental Material for: PHYLOGENY, SCALING, AND THE GENERATION OF EXTREME FORCES IN TRAP-JAW ANTS JOSEPH C. SPAGNA, ANTONIS I. VAKIS, CHRIS A. SCHMIDT, SHEILA N. PATEK, XUDONG ZHANG, and ANDREW V. SUAREZ

The images, within their original folders (e.g. 093004as4), need to be placed in a directory called 'Images,' located in the current working directory So far, all captures have been saved as sequential TIFF files First, run the tracing program

The folder name is saved as a parameter called: filename

Specify the number of frames per second (fps)

Specify the width of video in pixels (width)

Specify the height of video in pixels (height)

Specify the image resolution in dots per inch (dpi)

Viewing the image thumbnails, select the range of images over which the full motion of the mandibles is captured

Specify whether the strike involves both or just one of the mandibles

First, select the approximate location of the mandible base

Then, sequentially select the left and right mandible tips over the selected frames of interest

Finally, reselect the approximate location of the mandible base; an average mandible position will be used for the calculations

In the case where it is unknown, the magnification can be calculated from the average recorded mandible length for each species Magnification parameter: zoom Default value = 10x

Finally, run the calculations program The name of the folder must be specified; this is the same name as the .mat file to be used

Calculations: Rcrude Mandible base, A (x1, y1) Rcrude Closed position B (x2, y2)

Calculations: Grid • Select number of grid points/ possible positions of centers of rotation e.g. 40 intervals, 41 x 41 = 1,681 grid points • Create grid centered around the calculated mandible base location cor_values_min_x = base_coords(1,1) – Rcrude cor_values_max_x = base_coords(1,1) + Rcrude cor_values_min_y = base_coords(1,2) – Rcrude cor_values_max_y = base_coords(1,2) + Rcrude

Calculations: Grid • Create a matrix with grid coordinates using values calculated for the minima and maxima in the x and y directions, as well as the interval • Reshape the resulting 41 x 41 x 2 matrix to a matrix of size 1681 x 2 (cor_values matrix)

Calculations: Rvalues • Having established the number of iterations for the left and right centers of rotation locations, specify the number of iterations for each mandible radius (values, e.g. 200) • Specify: Rmin = Rcrude – Rcrude/2 (50% smaller) Rmax = Rcrude + Rcrude/2 (50% larger) • Rvalues = linspace(Rmin,Rmax,values)

Calculations: Iterations • Using nested for loops, iterate the values of x, y and R for the left and right mandibles for # of possible radius values (values) (p) for # of elements (41 x 41 = 1,681) (k) for # of frames (e.g. 8) (i) to get the RMS_error for each mandible

Calculations: RMSerror • The RMS (Root Mean Square) error is calculated from the formula *: where radius is defined as: and 3 is the number of free parameters (x, y, R) for each mandible * Methodology of circle fitting using RMS error adapted from Amara Graps of Stanford University’s Solar Oscillation Investigations (http://www.mathcom.com/corpdir/techinfo.mdir/scifaq/q230.html)

Calculations: Minimum RMSerror • The resulting matrices of the left and right mandible RMS_error values are reshaped into two new matrices of size 336,200 x 1, where 336,200 = 41 x 41 x 200 • Using a simple algorithm (min command), the minimum values of the RMS_error are found (RMSleft_min & RMSright_min) • These are used to extract the values of x, y and R where the error is minimum for each mandible

Calculations: RMSerror plots

Calculations: Radii & c.o.r.

Calculations: ω and α • To calculate the angular velocities and accelerations use the following formulae, based on the difference of the slopes in each frame: θ(i) = atan [coords(i,2) – x0y0(1,2)] [coords(i,1) – x0y0(1,1)] ω(i) = [θ(i+1) – θ(i)]*fps (rad/s) α(j) = [ω(j+1) – ω(j)]*fps (rad/s/s) where θ(i) is the slope of each frame, ω(i) is the ang. velocity, α(j) is the ang. acceleration, and fps are the frames per sec. For n frames, there are n-1 velocity and n-2 acceleration values

Calculations: u and a • In order to convert the angular velocities and accelerations into their linear counterparts, the radius of each mandible is needed in meters: Rnew = (R/ dpi/ zoom)*0.0254 (m) • Then, the linear velocities and accelerations are: u(i) = ω(i)*Rnew (m/s) a(j) = α(j)*Rnew (m/s/s)

Calculations: Velocity Plots

Calculations: Acceleration Plots

Output of Calculations Program

MATLAB Mandible Tracing & Calculations Program