> For the complete documentation index, see [llms.txt](https://gitbook.saf.guide/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://gitbook.saf.guide/getting-started/geometry.md).
# 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](https://saf.guide/Content/A_Objects/5_StructuralPointConnection.htm).
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 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
|
## Notes
{% hint style="info" %}
**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:**
{% endhint %}
