Sum type of all geometric surface shapes.

Arbitrary sphere \[ F(x,y,z) = (x-P_x)^2 + (y-P_y)^2 + (z-P_z)^2 - R^2 = 0 \]

Sphere with center on x-axis \[ F(x,y,z) = (x-P_x)^2 + y^2 + z^2 - R^2 = 0\]

Sphere with center on y-axis \[ F(x,y,z) = x^2 + (y-P_y)^2 + z^2 - R^2 = 0\]

Sphere with center on z-axis \[ F(x,y,z) = x^2 + y^2 + (z-P_z)^2 - R^2 = 0\]

Arbitrary plane \[ F(x,y,z) = A x + B y + C z - D = 0 \]

Plane with normal vector in direction of x-axis. \[ F(x,y,z) = x - D = 0 \]

Plane with normal vector in direction of y-axis. \[ F(x,y,z) = y - D = 0\]

Plane with normal vector in direction of z-axis. \[ F(x,y,z) = z - D = 0 \]

Arbitrary cylinder with radius \( R \) and central line defined by a position vector \( P = (P_x,P_y,P_z)^T \) and a direction vector \( D = (D_x,D_y,D_z)^T \). \[ F(x,y,z) = ||X-P||_2^2 - \langle X-P,D_n\rangle^2 - R^2 = 0 \] where: \[X = \begin{pmatrix}x\\y\\z\end{pmatrix}\] \[ D_n = \frac{D}{||D||_2} \]

Cylinder with radius \( R \) and x-axis as central line. \[ F(x,y,z) = y^2 + z^2 - R^2 = 0\]

Cylinder with radius \( R \) and y-axis as central line. \[ F(x,y,z) = x^2 + z^2 - R^2 = 0\]

Cylinder with radius \( R \) and z-axis as central line. \[ F(x,y,z) = x^2 + y^2 - R^2 = 0\]

Cylinder with radius \( R \) parallel to x-axis and central going through \( (0,P_y,P_z)^T \) \[ F(x,y,z) = (y-P_y)^2 + (z-P_z)^2 - R^2 = 0\]

Cylinder with radius \( R \) parallel to y-axis and central going through \( (P_x,0,P_z)^T \) \[ F(x,y,z) = (x-P_x)^2 + (z-P_z)^2 - R^2 = 0\]

Cylinder with radius \( R \) parallel to z-axis and central going through \( (P_x,P_y,0)^T \) \[ F(x,y,z) = (x-P_x)^2 + (y-P_y)^2 - R^2 = 0\]

Paraboloid of revolution with x-axis as central line opened in positive direction and its vertex at \( (P_x,0,0)^T \). \[ F(x,y,z) = -(x-P_x) + \frac{y^2}{A^2} + \frac{z^2}{A^2} = 0\]

Paraboloid of revolution with y-axis as central line opened in positive direction and its vertex at \( (0,P_y,0)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} - (y-P_y) + \frac{z^2}{A^2} = 0\]

Paraboloid of revolution with z-axis as central line opened in positive direction and its vertex at \( (0,0,P_z)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} + \frac{y^2}{A^2} - (z-P_z) = 0\]

Paraboloid of revolution opened in positive x-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = -(x-P_x) + \frac{(y-P_y)^2}{A^2} + \frac{(z-P_z)^2}{A^2} = 0\]

Paraboloid of revolution opened in positive y-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} -(y-P_y) + \frac{(z-P_z)^2}{A^2} = 0\]

Paraboloid of revolution opened in positive z-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} + \frac{(y-P_y)^2}{A^2} - (z-P_z) = 0\]

Paraboloid of revolution with x-axis as central line opened in negative direction and its vertex at \( (P_x,0,0)^T \). \[ F(x,y,z) = (x-P_x) + \frac{y^2}{A^2} + \frac{z^2}{A^2} = 0\]

Paraboloid of revolution with y-axis as central line opened in negative direction and its vertex at \( (0,P_y,0)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} + (y-P_y) + \frac{z^2}{A^2} = 0\]

Paraboloid of revolution with z-axis as central line opened in negative direction and its vertex at \( (0,0,P_z)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} + \frac{y^2}{A^2} + (z-P_z) = 0\]

Paraboloid of revolution opened in negative x-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = (x-P_x) + \frac{(y-P_y)^2}{A^2} + \frac{(z-P_z)^2}{A^2} = 0\]

Paraboloid of revolution opened in negative y-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} + (y-P_y) + \frac{(z-P_z)^2}{A^2} = 0\]

Paraboloid of revolution opened in negative z-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} + \frac{(y-P_y)^2}{A^2} + (z-P_z) = 0\]

Cone of revolution with x-axis as central line opened in positive direction and its vertex at \( (P_x,0,0)^T \). \[ F(x,y,z) = - sign(x-P_x)\cdot(x-P_x)^2 + \frac{y^2}{A^2} + \frac{z^2}{A^2} = 0\]

Cone of revolution with y-axis as central line opened in positive direction and its vertex at \( (0,P_y,0)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} - sign(y-P_y)\cdot(y-P_y)^2 + \frac{z^2}{A^2} = 0\]

Cone of revolution with z-axis as central line opened in positive direction and its vertex at \( (0,0,P_z)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} + \frac{y^2}{A^2} - sign(z-P_z)\cdot(z-P_z)^2 = 0\]

Cone of revolution opened in positive x-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = -sign(x-P_x)\cdot(x-P_x)^2 + \frac{(y-P_y)^2}{A^2} + \frac{(z-P_z)^2}{A^2} = 0\]

Cone of revolution opened in positive y-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} -sign(y-P_y)\cdot(y-P_y)^2 + \frac{(z-P_z)^2}{A^2} = 0\]

Cone of revolution opened in positive z-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} + \frac{(y-P_y)^2}{A^2} - sign(z-P_z)\cdot(z-P_z)^2 = 0\]

Cone of revolution with x-axis as central line opened in negative direction and its vertex at \( (P_x,0,0)^T \). \[ F(x,y,z) = sign(x-P_x)\cdot(x-P_x)^2 + \frac{y^2}{A^2} + \frac{z^2}{A^2} = 0\]

Cone of revolution with y-axis as central line opened in negative direction and its vertex at \( (0,P_y,0)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} + sign(y-P_y)\cdot(y-P_y)^2 + \frac{z^2}{A^2} = 0\]

Cone of revolution with z-axis as central line opened in negative direction and its vertex at \( (0,0,P_z)^T \). \[ F(x,y,z) = \frac{x^2}{A^2} + \frac{y^2}{A^2} + sign(z-P_z)\cdot(z-P_z)^2 = 0\]

Cone of revolution opened in negative x-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = sign(x-P_x)\cdot(x-P_x)^2 + \frac{(y-P_y)^2}{A^2} + \frac{(z-P_z)^2}{A^2} = 0\]

Cone of revolution opened in negative y-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} + sign(y-P_y)\cdot(y-P_y)^2 + \frac{(z-P_z)^2}{A^2} = 0\]

Cone of revolution opened in negative z-direction with its vertex at \( (P_x,P_y,P_z)^T \) \[ F(x,y,z) = \frac{(x-P_x)^2}{A^2} + \frac{(y-P_y)^2}{A^2} + sign(z-P_z)\cdot(z-P_z)^2 = 0\]

Torus of revolution in x-direction with its center at \( (P_x,P_y,P_z)^T \). \[ F(x,y,z) = \left( \sqrt{(y-P_y)^2 + (z-P_z)^2} - R_2 \right)^2 + (x-P_x)^2 - R_1^2 = 0\]

Torus of revolution in z-direction with its center at \( (P_x,P_y,P_z)^T \). \[ F(x,y,z) = \left( \sqrt{(x-P_x)^2 + (z-P_z)^2} - R_2 \right)^2 + (y-P_y)^2 - R_1^2 = 0\]

Torus of revolution in z-direction with its center at \( (P_x,P_y,P_z)^T \). \[ F(x,y,z) = \left( \sqrt{(x-P_x)^2 + (y-P_y)^2} - R_2 \right)^2 + (z-P_z)^2 - R_1^2 = 0\]

3x2 Matrix with minimum and maximum coordinates in each dimension.

Type for a geometric ray wich has an origin and a direction. To distinguish this type from an optical ray, it is referred to as a halfline or simply a line.

Datatype for the orientation (direction) of an axis.

Datatype for ascending or descending order.

Datatype for the unit of an angle which could either be radiant or degrees.

The Datatype System represents the whole optical system.

A Part is a physical object with a geometric shell and a material that light can interact with.

Datatype for an optical surface. It is composed as the geometric shape of the surface together with an flag that defines if the shape is inverted. A surface can also have optical properties.

A Material is defined by it's physical properties that are relevant for optics.

Various ideal materials can be modelled with the following values for refractive index and internal transmittance. Please note that this is a simplification since there is also a complex refractive index, but it works for refraction based on Snell's model:

• ideal mirrors : mRefract = Refract 0.
• ideal absorbers : mRefract = Refract infinity ; mTransmit = Transmit 0
• ideal transmitters : mTransmit = Transmit 1

Datatype for different methods of how to determine the refractive index

Catalog with named Sellmeier coefficients for different materials which is defined as a list of pairs. Each pair consists of a material name as a String and the corresponding Sellmeier coefficients. Please see the examples on how to use this.

The Sellmeier Coefficients are used to implement dispersion, that is the refractive index is dependent of the wavelength. \[n^2(\lambda) = 1 + \frac{B_1\lambda^2}{\lambda^2-C_1} + \frac{B_2\lambda^2}{\lambda^2-C_2} + \frac{B_3\lambda^2}{\lambda^2-C_3}\]

The value for internal transmittance \( T_i \) of a material is between 0 and 1 and describes the preservation of irradiance in a unit depth \(d\) of 1cm. Info: There is a relashionship to the materials absorption coefficient \( k \) \[ T_i = \exp^{-k\cdot d} \]

Method that determines the behaviour of raytracing

An ray is defined by its geometric properties orgin and direction together with physical properties.

Datatype for the result after raytracing.

Where a ray hits a part there is an intersection. This datatype describes all the geometric and optical properties of an intersection.

Creates a 2d-cone of lines in the xy-plane with the same origin (0,0,0) and central direction (1,0,0).

Parameters:

• Double : aperture angle of the lines in degrees
• Int : number of lines
• [Line] : resulting lines

Example:

lines = angularLines2d 15 10

Available in: Free | Base | Pro

Creates a cone of lines in 3d with the same origin (0,0,0) and central direction (1,0,0).

Parameters:

• Double : aperture angle of the lines in degrees
• Int : number of lines
• [Line] : resulting lines

Example:

lines = angularLines3d 15 200

Available in: Free | Base | Pro

Creates a list of parallel lines in yz-plane in positve x-direction. The origin of the beam center is in (0,0,0).

Parameters:

• Double : lines width
• Int : number of lines
• [Line] : resulting lines

Available in: Free | Base | Pro

Creates a list of parallel lines with circular profile in positive x-direction. The orgin of the beam center is in (0,0,0).

Parameters:

• Double : lines width (diameter)
• Int : number of lines
• [Line] : resulting lines

Available in: Free | Base | Pro

6-Parameter Transformation of a Line. This function can be used to move a newly created line to some arbitrary orientation and position.

Parameters:

• V3 Double : euler angles (alpha, beta, gamma) in degrees of the rotation matrix for the new orientation. The order of rotation is:

  1. Rotate around z-Axis with angle alpha
  2. Rotate around x-Axis with angle beta
  3. Rotate around z-Axis with angle gamma
• V3 Double : translation vector (tx, ty, tz) to the new position
• Line : line to transform
• Line : resulting line

Available in: Free | Base | Pro

6-Parameter Transformation of multiple Lines. The transformation parameters are the same as in moveLine.

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Infinity is larger than any real number. Here it is defined as 1/0.

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Epsilon is the approximation to machine epsilon, which is at 2.220446049250313e-16 for double precision (64-bit). This value can be used in direct or derived form (depending on the context) for algorithms that require a small positive tolerance.

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Calculates the diameter of a box.

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This Function is for trace a single ray through a system. The raytrace ends when the source ray or any deflected ray of the trace has no more next intersection. This also means if a ray does not hit any surface, it won't be added to the raytrace and therefore is not visualized. To visualize all possible rays, you can add a part which is a bounding box around all the other parts of the system. This way there is an intersection where an escaping ray hits the bounding box and it will be visualized. The bounding box should be of the same material as the surrounding to ensure that there is no deflection.

Available in: Free | Base | Pro

Raytracing of multiple rays through a system. The same rules as are described in traceRay apply to each single ray.

Available in: Free | Base | Pro

Returns the list of parts where a point lies inside.

Available in: Base | Pro

Tests if a point lies inside an part.

Parameters:

• Part : part
• V3 Double : point
• Bool : boolean return value

Available in: Base | Pro

Converts lines in a bulk to rays by assigning them all the same optical properties. Be careful with this function, since all line origins should lie within parts of the same material that is assigned.

Parameters:

• Double : wavelength of the rays light in vacuum in micrometers
• Material : assigned material where the orgins lie in
• [Line] : list of halflines
• [Ray] : list of resulting rays

Available in: Free | Base | Pro

Exports parts in various formats for import into external CAD programs. The file format is identified by the extension in the filepath. The following formats are supported:

• XYZ (extension .xyz): Sampled 3d points on the part shells are saved in a text file. Each line consists of the coordinates of one point separated by a whitespace. There is no separation between different parts in the exported file.
• STEP (extension .stp or .step): Parts are exported in their boundary representation. The part surfaces are stored in analytical form and the edges at surface intersections are stored as precisely calculated spline curves. A resolution around 0.001 times the bounding box diameter usually works fine. It can be decreased for parts with closely adjacent vertices or edges, but there are limitations due to longer computation times. You can also use a tighter bounding box on particular parts and export them separately. Please note that some CAD softwares can have problems with the STEP import. It was observed that sometimes the wrong side of a surface is clipped or revolved surfaces such as paraboloids are missing. There are two testfiles 'TestParts.stp' and 'FirstLens.stp' in the examples folder to test the import capabilities of other programs.

Parameters:

• FilePath : output filepath, a relative path is possible
• Double : resolution of a point sampling on the parts shells
• Box : bounding box that encloses the parts
• [Part] : parts to export
• IO () : returned IO action

Available in: Base | Pro

Before visualizing any graphical content a view object must be initialized with initView. It's possible to create multiple view instances. The view coordinate system is such that the coordinate origin (0,0,0) lies in the center of the screen and the x-axis points to the right and the y-axis to the top. The z-axis points upward from the screen into the users eye. Call this function within an IO block.

Parameters:

• String : window title
• V2 Int : window size
• V2 Int : window position
• Box : bounding box of the volume to render
• IO View : returned IO action with the view

Example:

view <- initView "myTitle" (800,600) (100,100) ((20,20),(20,20),(20,20))

Available in: Free | Base | Pro

Shows a view and runs its renderloop. Call this function within an IO block. During the renderloop the following user interactions could be triggerd with mouse and keyboard events:

• pan: Press the right mouse button and move the cursor
• zoom: Press the right mouse button and scroll mouse-wheel
• xy-Rotation: Press left mouse button and move the cursor
• z-Rotation: Move the mouse cursor to the center of rotation, press the left and right mouse button, move the cursor away from the center and then sideways
• redraw: Some graphic content, such as the parts, is rendered as a pointcloud. This pointcloud can be recalculated and redrawed for different sights after a navigation. You can trigger a redraw with the Space key
• close: Close the view window with the Esc key

Available in: Free | Base | Pro

Shows multiple views and runs their mainloops. User interactions are accepted for each of them. Call this function within an IO block.

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Adds parts to a view. Call this function within an IO block.

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Adds a single point to a view. Call this function within an IO block.

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Adds multiple points to a view. Call this function within an IO block.

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Adds a single line to a view. Call this function within an IO block.

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Adds multiple lines to a view. Call this function within an IO block.

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Adds a single ray to a view. Call this function within an IO block.

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Adds multiple rays to a view. Call this function within an IO block.

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Adds a single raytrace to a view. Call this function within an IO block.

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Adds multiple raytraces to a view. Call this function within an IO block.

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