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3.3. Element Types and Affine Equivalent Triangulations 119

Here, in the ith column a i has been replaced with x. Since in general,
ˆ
vol (K)= vol (K) | det(B)| (3.50)
ˆ
for the reference simplex K defined by (3.47) (cf. (2.50)), we have for the
volume of the d-simplex K =conv {a 1 ,... ,a d+1 },

 

a 11 ··· a 1,d+1

 . . . 
1
 . . . . . . 
vol (K)=
det  
,
d!
 
a d1 ··· a d,d+1

1 ··· 1

and from this,
vol (conv {a 1 ,...,x,...,a d+1})
λ i (x)= ± . (3.51)
vol (conv {a 1 ,...,a i ,...,a d+1})
The sign is determined by the arrangement of the coordinates.
In the case d =2 for example, we have
vol(K)= det(B)/2
⇐⇒ a 1 ,a 2 ,a 3 are ordered positively (that is, counterclockwise).
Here, conv {a 1 ,... ,x,...,a d+1 } is the d-simplex that is generated by re-
placing a i with x and is possibly degenerate if x lies on a face of K (then
λ i (x) = 0). Hence, in the case d =2we have for x ∈ K that the barycentric
coordinates λ i (x) are the relative areas of the triangles that are spanned by
x and the vertices other than a i . Therefore, we also speak of surface coordi-
nates (see Figure 3.4). Analogous interpretations hold for d =3. Using the
barycentric coordinates, we can now easily specify points that admit a ge-
ometric characterization. The midpoint a ij := 1 (a i + a j ) of a line segment
2
that is given by a i and a j satisfies, for instance,
1
λ i (x)= λ j (x)= .
2
By the barycentre of a d-simplex we mean
d+1
1 1
a S := a i ;thus λ i (a S )= for all i =1,... ,d +1 . (3.52)
d +1 d +1
i=1
A geometric interpretation follows directly from the above considerations.
In the following suppose conv {a 1 ,...,a d+1} to be a regular d-simplex.
We make the following definition:
Finite Element: Linear Ansatz on the Simplex

K =conv {a 1 ,...,a d+1} ,
P = P 1 (K) , (3.53)
Σ = {p (a i ) ,i =1,... ,d +1} .