Plane Angle and In-Plane Movement Functions

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# Pitch_Plane_Functions.r #
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# The following contains several functions related to the plane of a pitch using PITCHf/x data and movement.

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# Binormal Vector #
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# PITCH is a list containing the 9 PITCHf/x parameters

Binormal_Vector <- function(PITCH){
    # Set the velocity vector
    V <- c(PITCH$vx0,PITCH$vy0,PITCH$vz0);
    # Set the acceleration vector
    A <- c(PITCH$ax,PITCH$ay,PITCH$az);
    # Find and normalize the binormal vector
    V_x_A <- c(V[2]*A[3] - V[3]*A[2],V[3]*A[1] - V[1]*A[3],V[1]*A[2] - V[2]*A[1]);
    V_x_A_norm <- sqrt(V_x_A[1]^2 + V_x_A[2]^2 + V_x_A[3]^2);
    return(V_x_A/V_x_A_norm);
}

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# PITCHf/x Basis #
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# PITCH is a list containing the 9 PITCHf/x parameters

PFX_Basis <- function(PITCH){
    # Set the binormal vector
    V1 <- Binormal_Vector(PITCH);
    # Set the u-vector
    V2 <- sign(V1[1])*c(-V1[2],V1[1],0)/sqrt(V1[1]^2 + V1[2]^2);
    # Set the w-vector
    V3 <- sign(V1[1]*V2[2]-V1[2]*V2[1])*c(-V1[3]*V2[2],V1[3]*V2[1],V1[1]*V2[2]-V1[2]*V2[1]);
    V3 <- V3/sqrt(V3[1]^2 + V3[2]^2 + V3[3]^2);
    # Build a matrix containing the vectors
    V <- array(0,dim=c(3,3));
    V[,1] <- V1;
    V[,2] <- V2;
    V[,3] <- V3;
    return(V);
}

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# Angle of Binormal Vector from Vertical #
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# PITCH is a list containing the 9 PITCHf/x parameters

Vertical_Angle <- function(PITCH){
    # Find the PITCHf/x basis
    BV <- PFX_Basis(PITCH);
    BV_T <- t(BV);
    # Set the w-vector
    W <- BV_T[3,];
    # Find the angle w makes with vertical (0,0,1)
    rho <- 180*acos(abs(W[3]))/pi;
    # Assign the angle the correct sign
    if (W[1] < 0){rho <- -abs(rho);} else {rho <- abs(rho);}
    return(rho);
}

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# In-Plane Movement #
#####################

# PITCH is a list containing the 9 PITCHf/x parameters

IP_Movement <- function(PITCH){
    # Find the time for the pitch to reach 40 feet in xyz-space
    a <- 0.5*PITCH$ay;
    b <- PITCH$vy0;
    c_40 <- PITCH$y0 - 40;
    t_40 <- (-b - sqrt(b^2 - 4*a*c_40))/(2*a);
    # Find the time for the pitch to reach the plate in xyz-space
    c_plate <- PITCH$y0 - (17/12);
    t_plate <- (-b - sqrt(b^2 - 4*a*c_plate))/(2*a);
    # Find the PITCHf/x basis
    BV <- PFX_Basis(PITCH);
    BV_T <- t(BV);
    # Find the w-acceleration of the pitch and w-component of gravity
    aw <- BV_T[3,] %*% c(PITCH$ax,PITCH$ay,PITCH$az);
    gw <- BV_T[3,] %*% c(0,0,-32.174);
    # Compute the in-plane movement, in inches
    return(6*(aw-gw)*(t_plate - t_40)^2);
}

##################################
# In-Plane Movement with Gravity #
##################################

# PITCH is a list containing the 9 PITCHf/x parameters

IP_Movement_G <- function(PITCH){
    # Find the time for the pitch to reach 40 feet in xyz-space
    a <- 0.5*PITCH$ay;
    b <- PITCH$vy0;
    c_40 <- PITCH$y0 - 40;
    t_40 <- (-b - sqrt(b^2 - 4*a*c_40))/(2*a);
    # Find the time for the pitch to reach the plate in xyz-space
    c_plate <- PITCH$y0 - (17/12);
    t_plate <- (-b - sqrt(b^2 - 4*a*c_plate))/(2*a);
    # Find the PITCHf/x basis
    BV <- PFX_Basis(PITCH);
    BV_T <- t(BV);
    # Find the w-acceleration of the pitch
    aw <- BV_T[3,] %*% c(PITCH$ax,PITCH$ay,PITCH$az);
    # Compute the in-plane movement, in inches
    return(6*aw*(t_plate - t_40)^2);
}