. Author manuscript; available in PMC: 2017 Aug 1.

Published in final edited form as: Magn Reson Med. 2015 Oct 19;76(2):663–678. doi: 10.1002/mrm.25888

Table 1.

Symbols

Symbol

Definition

Fourier Domain MR signal

k-space vector

2D polygon

\in R^{2}

E

Total number of vertices or edges

e

index of e^th vertex or edge

V_{e}

e^th vertex, first and last vectors connected,
i.e.

V_{E + 1} = V_{1}

and

V_{E} = V_{0}

r^{(V_{e})}

position vector associated vertex

V_{e}

L_{e}

L_{e} = r^{(V_{e + 1})} - r^{(V_{e})}

,
vector oriented in the direction the e^th edge, pointed from vertex

V_{e}

V_{e + 1}

with the same length as the edge

{\hat{t}}_{e}

{\hat{t}}_{e} = L_{e} / L_{e}

,
unit vector oriented in the direction of the e^th edge

{\hat{n}}_{e}

is a unit vector normal to the e^th edge and pointing outward from the polygon

r^{(C_{e})}

r^{(C_{e})} = r^{(V_{e})} + {\hat{t}}_{e} L_{e} / 2

, position vector of the midpoint between vertices

V_{e}

and

V_{e + 1}

Area of Polygon

3D polyhedron

\in R^{3}

F

total number of faces in the polyhedron

f

index of f^th face

{\hat{N}}_{f}

outward normal of the f^th face

E_{f}

total number of edges or vertices of the f^th face

V_{fe}

e^th vertex of f^th face, first and last vectors connected,
i.e.

V_{f, E_{f} + 1} = V_{f 1}

and

V_{f E_{f}} = V_{f 0}

r^{(V_{fe})}

position vector associated vertex

V_{fe}

L_{fe}

L_{fe} = r^{(V_{f, e + 1})} - r^{(V_{fe})}

,
vector oriented in the direction the e^th edge, pointed from vertex

V_{fe}

V_{f, e + 1}

with the same length as the edge

{\hat{t}}_{fe}

{\hat{t}}_{fe} = L_{fe} / L_{fe}

,
unit vector in the direction of the e^th edge of the f^th face

{\hat{n}}_{fe}

outward normal to the e^th edge of the f^th face and is in the plane of the face

r^{(C_{fe})}

r^{(C_{fe})} = r^{(V_{fe})} + {\hat{t}}_{fe} L_{fe} / 2

, position vector of the midpoint between vertices

V_{fe}

and

V_{f, e + 1}

P_{f}

Area of f^th face

Volume of Polyhedron