Geometry

Definition of Geometries in KostaCLOUD Software Suite

Geometry in KostaCLOUD is defined in a very generic way where volumes are referenced via a global reference point with a reference axis. As shown below, when an object is clicked its local reference coordinate system is shown. The circle with the "1" inside of it is the origin point for this reference coordinate system. It provides generic referencing for objects. The difference axes emanating from this point are the global coordinate axes. The origin sphere can be clicked and dragged to move the object within the view plane. The colored squares can be clicked to move the object with a given reference plane, such as YZ plane in this case. The quarter circle lines can be clicked and dragged to rotate your object in 15° increments perpendicular to the plane they are drawn in. The arrow tips can be clicked and dragged to move an object in a line exactly in the axis/direction they define.

Sphere with Local Reference

Each point in these objects is transformed using a transformation matrix as shown below:

$$\begin{bmatrix} x_g\\ y_g\\ z_g \end{bmatrix} = \begin{bmatrix} a_{xx} & a_{xy} & a_{xz}\\ a_{yx} & a_{yy} & a_{yz}\\ a_{zx} & a_{zy} & a_{zz} \end{bmatrix} \begin{bmatrix} x_L\\ y_L\\ z_L \end{bmatrix} + \begin{bmatrix} x_0\\ y_0\\ z_0 \end{bmatrix}$$

Where the 3x3 matrix made up of rotation matrix elements with local coordinate vector multiplying it and the rightmost vector as the global position of the local origin.

The rotation matrix is calculated by three matrix multiplications in the ZYX direction where:

$$\begin{bmatrix} a_{xx} & a_{xy} & a_{xz}\\ a_{yx} & a_{yy} & a_{yz}\\ a_{zx} & a_{zy} & a_{zz} \end{bmatrix} = \begin{bmatrix} \cos\theta\cos\phi & \cos\theta\sin\phi\sin\psi-\sin\theta\sin\phi & \cos\theta\sin\phi\cos\psi+\sin\theta\sin\psi\\ \sin\theta\cos\phi& \sin\theta\sin\phi\sin\psi+\cos\theta\cos\psi& \sin\theta\sin\phi\cos\psi-\cos\theta\sin\psi\\ -\sin\phi & \cos\phi\sin\psi & \cos\phi\cos\psi \end{bmatrix}$$

Additionally all Geometric elements have intrinsic properties. Each has the ability to "Absorb" "Reflect" or represent a "Material". There are a few element types where these options are no available such as for Paraxial Elements, Reference Surfaces and Apertures. If an element represents a material there is an option to ignore this material. Materials can be searched for via the Material database search, where global material databases are available as well as local material databases.