Geometry

# Geometry

Basic construction elements are simple geometry types, that are used for shape definition of structural members and geometrical object. With defining the further attributes to these elements from the lists Structural analysis elements, Supports and hinges and Loads the complete structural model is created. All values refer to the list StrucutralPointConnection.
Following geometrical types are available:
Text
Geometry type
Type definition
Insertion data explanation
SAF geometry strings
Notes Line
Straight-line between two nodes
Start point , End point
N1;N2
Line
- Circular Arc
Arch defined with 3 nodes
Start point, Intermediate point, End point
N3;N4;N5
Circular Arc
- Parabolic Arc
Parabolic arch defined with 3 nodes
Start point, Intermediate point, End point
N6;N7;N8
Parabolic Arc
- Bezier
Cubic Bezier curve
Start point, 2nd point of control polygon (vertex), 3rd point of control polygon (vertex), End point
N9;Vertex_B1_1;
Vertex_B1_2;N10
Bezier
N9 and N10 stands for start and end node
Vertex_B1_1, Vertex_B1_2 define vertexes of bezier curve
All values refers to list StrucutralPointConnection
Bezier curve is parabolic, when 2nd and 3rd control points are the identical (values of coordinates are the same) Spline
Curved line defined by polynomial function
Start point, Set of mid points, End point
N11;N12;N13;N14;N15;N16;N17;N18
Spline-8
"Spline-#" where "#" stands for number of nodes defining the spline Circle
Circle
Center Point, Point on the perimeter
N36;N37
Or
Three point on perimeter
N36;N37;N38
Circle and Point
or
Circle by 3 points
Circle is not valid to define StrucutralCurveMember Polyline
Combination of in nodes connected geometric types
List of nodes
N21;N22;N23;N24;N25;N26;N27;N28;N29; N30;N31;N32;N33;Vertex_B1_1;VertexB_1_2;N34;N35
Line;Line;Spline-7;Line;Circular Arc;Line;Bezier;Line
Detail explanation can be found in notes below
Mathematical definitions:
• Bezier
$Q(t)=\sum_{i=0}^{3}P_iB_i(t)$
;
$t\epsilon<0,1>$
$B_{0t}=(1-t)^3,B_{1t}=3t(1-t)^2,B_{2t}=3t^2(1-t),B_{3t}=t^3$
$Q(t)$
is for the Bezier curve
$P_{i}$
is for coodinates, and
$B_{it}$
is for basis function \
• Spline
$Q(t)=\sum_{i=0}^{m}P_iN_i^n(t)$
;
$t\epsilon$
$N_i^0(t)=1$
for
$t\epsilon$
$N_i^0(t)=0$
otherwise
$N{i}^{k}{(t)}=\frac{t-t_i}{t_{i+k}-t_i}N_{i}^{k-1}{(t)}+\frac{t_{i+k+1}-t}{t_{i+k+1}-t_{i+1}}N_{i+1}^{k-1}{(t)}$
$Q(t)$
is for the Spline curve
$P_i$
is for the coordinates
$N_i^n(t)$
is for basis function
$n$
is for the degree of curve
$m$
is for points of the control polygon
Polyline schematics: 1