imbalSubdiv_A {treeDbalance}

Calculation of the centroid angles

Description

imbalSubdiv_A - Calculates the node imbalance value "centroid angle" of a vertex which subdivides the edge (p,v) at v+x \cdot (p-v) with x \in [0,1]. For example, we can obtain the node imbalance value of v if x=0, and x=0.5 would indicate a subdividing node exactly in the middle of v and p.
Attention: If x=1, this function will not calculate the node imbalance value of p with respect to its incoming edge but with respect to the edge (p,v) itself. This enables us to estimate the node imbalance integrals over the entire edge length.

imbalSubdiv_alpha - Calculates the node imbalance value "minimal centroid angle" of a vertex which subdivides the edge (p,v) at v+x \cdot (p-v) with x \in [0,1]. For example, we can obtain the node imbalance value of v if x=0, and x=0.5 would indicate a subdividing node exactly in the middle of v and p.
Attention: If x=1, this function will not calculate the node imbalance value of p with respect to its incoming edge but with respect to the edge (p,v) itself. This enables us to estimate the node imbalance integrals over the entire edge length.

angle3dVec - Calculates the angle in the interval [0,\pi] between two 3D vectors a and b. Note that the function returns 0 if one entry vector is (0,0,0).

Usage

imbalSubdiv_A(x, p, v, centr_v, centr_v_weight, edge_weight)

imbalSubdiv_alpha(x, p, v, centr_v, centr_v_weight, edge_weight)

angle3dVec(a, b)

Arguments

x

Numeric value \in [0,1] which indicates where on the edge (p,v) the subdivision takes place: v+x \cdot (p-v).

p

Numeric vector of size 3 (3D coordinates of parent node p).

v

Numeric vector of size 3 (3D coordinates of node v).

centr_v

Numeric vector of size 3 (3D coordinates of the centroid of the pending subtree of node v).

centr_v_weight

Numeric value >=0 (weight of the pending subtree of node v).

edge_weight

Numeric value >=0 (weight of the edge (p,v)).

a

Numeric vector of size 3 (e.g., 3D coordinates).

b

Numeric vector of size 3 (e.g., 3D coordinates).

Value

imbalSubdiv_A Numeric value \in [0,\pi] (higher values indicate a higher degree of asymmetry).

imbalSubdiv_alpha Numeric value \in [0,\pi/2] (higher values indicate a higher degree of asymmetry).

angle3dVec Numeric value in [0,\pi].

Author(s)

Sophie Kersting, Luise Kühn

Examples

imbalSubdiv_A(x=0.5,p=c(1,0,1),v=c(0,0,0),centr_v=c(0.5,0,0),
centr_v_weight=1,edge_weight=1)
imbalSubdiv_alpha(x=0.5,p=c(1,0,1),v=c(0,0,0),centr_v=c(0.5,0,0),
centr_v_weight=1,edge_weight=1)
angle3dVec(a=c(1,0,0),b=c(0,1,0)) # right angle = pi/2 = 1.5707...