# 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:

 ​ Geometry type Type definition Insertion data explanation SAF geometry strings Notes ​​ Line Straight-line between two nodes Start point , End pointN1;N2 Line - ​​ Circular Arc Arch defined with 3 nodes Start point, Intermediate point, End pointN3;N4;N5 Circular Arc - ​​ Parabolic Arc Parabolic arch defined with 3 nodes Start point, Intermediate point, End pointN6;N7;N8 Parabolic Arc - ​​ Bezier Cubic Bezier curve Start point, 2nd point of control polygon (vertex), 3rd point of control polygon (vertex), End pointN9;Vertex_B1_1;Vertex_B1_2;N10 Bezier N9 and N10 stands for start and end nodeVertex_B1_1, Vertex_B1_2 define vertexes of bezier curveAll values refers to list StrucutralPointConnectionBezier 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 pointN11;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 perimeterN36;N37OrThree point on perimeterN36;N37;N38 Circle and Point orCircle by 3 points Circle is not valid to define StrucutralCurveMember ​​ Polyline Combination of in nodes connected geometric types List of nodesN21;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

# Notes

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: