Table 2. Mathematical definition and meaning of the kinematic parameters implemented in the OpenCASA software.

Parameter	Mathematical definition	Meaning (from Mortimer [29])
Straight-line velocity (VSL)	$D\left(p_1,p_N\right)*\frac{FrameRate}{N-1}*\mathrm{\mu }$	VSL is determined by finding the straight-line distance between the first and last points of the trajectory and correcting for time. This value then gives the net space gain within the observation period
Curvilinear velocity (VCL)	$\left({\displaystyle \sum _{t=1}^{N-1}}D(p_t,p_{t+1})\right)*\frac{FrameRate}{N-1}*\mathrm{\mu }$	VCL is the distance travelled by the spermatozoon along its curvilinear path/s and is calculated by finding the sum of the distances along the trajectory then correcting for time. It refers to the total distance that the sperm head covers in the observation period
Average-path velocity (VAP)	$\left({\displaystyle \sum _{t=1}^{N-w}}D(q_t,q_{t+1})\right)*\frac{FrameRate}{N-w}*\mathrm{\mu }$	VAP is the distance the spermatozoon has traveled in the average direction of movement in the observation period. It is calculated by finding the length of the average path and correcting for time.
Linearity (LIN)	$\frac{VSL}{VCL}*100$	LIN is a comparison of the straight-line and curvilinear paths. It is an expression of the relationship between the two-dimensional projection of the three-dimensional path taken by the spermatozoon (i.e. curvilinear path) and its net space gain
Wobble (WOB)	$\frac{VAP}{VCL}*100$	WOB is the expression of the relationship between the average and curvilinear paths
Straightness (STR)	$\frac{VSL}{VAP}*100$	STR is a comparison of the straight-line and average paths and gives an indication of the relationship between the net space gain and the general trajectory of the spermatozoon
Amplitude of lateral head displacement ALHmean	Let the segment St(formed by two consecutive points between pt and pt+w−1) be the nearest segment to the segment S(qt,qt+1) relative to its middle point ${qm}_t=\frac{q_t+q_{t+1}}{2}$. ${ALH}_{mean}=2*\frac{{\sum }_{t=2}^{N-w}\mathrm{F}\left(D\right({qm}_t,S_t\left)\right)}{Count}*\mathrm{\mu }$ Where function F() and parameter Count are defined as: Global var Count = 0; // it counts the total number of elements that the function F() identify as local maxima. Function F(): If D(qmt,St)>D(qmt−1,St−1) and D(qmt,St)>D(qmt+1,St+1) then Count = Count+1 return D(qmt,St) else return 0	The amplitude of lateral head displacement (ALH) is used as an approximation of the flagellar beat envelope. It is not a true amplitude, in that it does not measure the perpendicular distance between the peak of a wave and the point of inflection of the curve, but rather gives the distance between the ‘peak’ and ‘trough’ of the centroid’s path. ALHmean is the mean of all of the ALH values along the trajectory.
ALHmax	Using the above definition: ${ALH}_{max}=2*{Max}_{t=2}^{N-w}\left[F\left(D\right({qm}_t,S_t\left)\right)\right]*\mathrm{\mu }$	ALHmax is the maximum ALH found along the trajectory.
Beat-cross frequency (BCF)	Let S(p1,p2) be a function that returns the segment made by the points p1 and p2. s1 and s2 being two different segments, let us define δ as $\delta (s_1,s_2)=\{\begin{array}{c}1if{s}_1\cap s_2\\ 0otherwise\end{array}$ Let $P={\left\{S(p_t,p_{t+1})\right\}}_{t=1}^{N-1}$ be the set of all consecutive segments that form a trajectory, and $Q={\left\{S(q_z,q_{z+1})\right\}}_{z=1}^{N-w}$ the set of all consecutive segments that form the corresponding average path for the trajectory {pt}. $BCF={\sum }_{t,z}\delta (s_p,s_q)*\frac{FrameRate}{N-w}$, with sp∈P,sq∈Q	BCF is the number of times the sperm head crosses the direction of movement, and this is related to the development of another flagellar wave
DANCE (DNC)	VCL*ALHmean	DNC is a measure of the pattern of sperm motion VCL×ALH
Mean angular displacement (MAD)	$\frac{{\sum }_{t=1}^{N-\Delta }\widehat{v_t}}{N-\Delta }$	MAD is a measure of the trajectory curvature, defined as ‘the time average of absolute values of the instantaneous turning angle of the head along its curvilinear trajectory’
Progressive Motility (PM)	Yes if STR > % and VAP> value, both defined by the user. Otherwise the trajectory is not considered progressive.	PM refers to sperm that are swimming in a mostly straight line
Motility trajectories	A trajectory is considered motile if the VCL parameter is greater than a value defined by the user (minimum vcl), and also the starting point of the trajectory is different than the last point.	This parameter refers to spermatozoa that are considered motile.
Fractal dimension (FD)	$FD=\frac{\mathrm{l}\mathrm{o}\mathrm{g}\left(n\right)}{\left[\log \left(n\right)+\mathrm{l}\mathrm{o}\mathrm{g}(\frac{d}{L})\right]}$ Where n is the number of track intervals (number of track points -1), d is the planar extent of the curve (maximum distance between the starting point and any other point of the track) and L is the length of the curvilinear path.	The fractal dimension is an expression of the degree to which a line fills a plane. It may be considered that the fractal dimension of a curve indicates its regularity. A curve with a low fractal dimension would be regular and predictable. Similarly, a curve with a high fractal dimension would have irregularly spaced changes in direction, apparently at random.

Let ${\left\{p_t\right\}}_{t=1}^N$ be a trajectory of length N defined as a sequence of N points; D(p,q) the Euclidean distance between the points p and q; and μ the scale factor (microns/pixel). Also let ${\left\{q_t\right\}}_{t=1}^{N-w+1}$ be the average trajectory of ${\left\{p_t\right\}}_{t=1}^N$ calculated applying a simple moving average with a rectangular window of size w ($q_t=\frac{1}{w}*{\sum }_t^{t+w-1}{p}_t$). Finally, let ${\widehat{v}}_t$ be the angle of the vector specified by the points ⟨p_t,p_t+Δ⟩, with Δ a positive integer lower than N.