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Key Sub-S ystem: Graphs In Shape graphs are used to display functions. The functions are either analytical or set interactively by adding control points to a curve. The control points of an interactive curve can also be loaded from an external ascii file. In modifiers the graphs are usually applied to set and display spatial variation of a quantity that is passed on to some quantity as a multiplier. In the modifiers the graphs may be single coordinate functions or they may depend on all three space coordinates in different types of coordinate systems (spherical by default, Cartesian or cylindrical). Generally the result is a single number, but in the case of the velocity field, for instance, the output can be a vector. In the Physics Module graphs commonly are a function of wavelength. The functionality and appearance of the graphs varies slightly depending on the particular context. The image of the graph on the right is taken from the Squeeze Modifier, which controls the shape of a cylindrically symmetric mesh. The function display at the top shows a line of how the function f(x) changes with the independent variable (always x). In this modifier it is the position along the axis of the object. At the bottom is a list of tabs to set a variety of f unction properties . Workflow: Analytic: By default the function is set to Point , but can be changed to Analytic with the Function drop-down list. In the analytic mode, the graph is controlled by a mathematical expression, that by default is the constant function f(x) = 1. Templates: From the Templates drop-down list a few commonly used functions can be selected that are the inserted as mathematical expressions in the editor line for the function. These templates can then be edited as needed. In our example we have typed the function manually. In additional to several standard mathematical operators such as "exp" and "abs", it includes the variables "a" and "b". Variables can have alphanumerical names. Reserved variables include "n", "t" and "Pi", referring to density, temperature and the number p. Variables: The numerical value of the variables is displayed in the Variables tab . Here the value can either be written manually or be assigned from the Math Module. If the variable was defined in the Math Module, then the flag labeled "Use global variables" needs to be set. If the flag is not set, the local manually set value is used (Local variable ). Global variables are then marked by a grey background. Global variables are accessible from any function throughout Shape. Constraints: Frequently used functions such as the Gaussian vary over an infinite range. In many practical situations the function is required to smoothly reach zero or some other fixed value within a finite range. This can be achieved by setting constraints in the Constraints tab. Here the function can be made to smoothly Fade in and Fade out within the ranges set there. The value form and to which the function converges is the "Default" which can be changed by the user. In the graph the region that is covert by the Fade in is shaded in blue and the Fade out in red. Point Graph: A point graph is set up using manually placed points. These point may also be loaded from a file (Load Ascii button). To obtain a continuous function the values between the points are interpolated. There are two modes for interpolation . By default the interpolation is linear between the points. The second is a spline interpolation that is controlled manually with separate handle on each side of the points. The spline interpolation is activated by clicking on the following icon: A green color in these icons at the top of the graph indicates that they are activated. To move the points in the graph use the left-click and drag mouse functionality. To move the spline handles , use Crtl- left-click and drag . Additional interactive functions to manipulate the points individually or collectively are available by activating some of the icons on the toolbar above the graph. If you hover over a button or icon, a tool tip shows a short description of the functionality, which are very self-explanatory in this case. The hand tool is active by default and allows one to move points by left-clicking on and dragging them. The magnifying glass: zoom in and out in the graph The next icon allows you to pan left-right and up-down in the graph. The box with a pencil is for selecting multiple points by dragging a rectangle around them, while the cursor symbols with the arrows activates moving the selected points together. Selected point are deleted by clicking on the cross icon . Note that unselected individual points can be added and removed from the right-click menu (see below). The range of the graph is adjust ed to the points in them by clicking on the icon with the two curvy arrows . As mentioned above the next icon activates the spline interpolation with two handles on each point. The wrench icon opens a pop-up windows that gives access to detailed options for the appearance of the graph . These should be largely self-explanatory. Right-click menu: The right-click menu give quick access to additional functionality for graphs. In particular there are four functions for point graphs at the top of the list. Add Point add a point to the curve where the curser is currently located. Remove Point removes a point over which the cursor is hovering. Set Point opens a small pop-up window where the exact x and y values of a point can be set. Mirror Point is a very useful function when a perfectly symmetric setting is required. When clicked while you hover over a point with coordinates (x,y) a copy of the point is added to the graph at (-x,y). The second set of functions toggle on and off various functions, which are self-explanatory. The Save Ascii function will open a dialog that allows you to save the point values of the graph in a file in ascii format. The Save Image function saves an image of the graph. In the file dialog that opens, the filename has to be give a suitable file extension for the image to be correctly saved. The extension can be one of typical file formats, such as .png or .jpg, etc. Finally, the Properties functions opens a dialog where the graph´s appearance may be customized in more detail by changing colors, tick-mark spacing, coordinate grids, etc. Special variants of graphs: Some contexts in Shape have specific features in addition to the basic functionality described above. These are explained in their specific contexts, such as the graph for the velocity modifier, which has a number of special functions. There is one common feature of the extended graphs that depend on more than one variable, which we describe here. Most function graphs have as output a single number that controls a modifier or displays some property. An exception is the velocity modifier, which has a vector as output. The basic graph that has been described above depends on a single variable. Many modifiers do, however, depend on more than one spatial dimension. Those have a separate graph that describes the spatial dependence of the output on each spatial coordinate. They are accessible through tabs at the top of the graph. The active coordinate is marked in grey. Using the default Custom Mode one can select from three different types of coordinate system in the Coordinates drop-down list: Spherical (default), Cartesian and Cylindrical. The labels of the access tabs for the graphs change their labels accordingly. Note that for the spherical coordinates the label convention is that of the North America, with q being longitude and f the latitude. SEPARABILITY: an important property of the default behavior of these graph is that the resulting function F(u,v,w) is separable in their component functions: F(u,v,w) = f0 * f(u) * f(v) * f(w) where f0 is a constant parameter set by the user under the Coordinates drop-down list. Non-separable functions: a non-separable analytic function can be set up when enabling the Analytic mode from the the Mode drop-down list. This feature requires an analytic description of the function and a point-graph can not be used. As before, one chooses the type of coordinate system from the Coordinates drop-down list. Then in the function editor a formula is types as a function of the general coordinates (u,v,w) as shown in the example on the right. The coordinates have to be in terms of the letters u,v and w, no matter which type of coordinates is chosen. Their meaning changes automatically to (x,y,z), (r, q,f) or (r, q, z ) for Cartesian, spherical or cylindrical coordinates, respectively. The image on the right shows a rendering of the density that was described by the analytic function above it. It illustrates the how the wavelength of the sinusoidal pattern can be continuously changed by mixing the coordinates appropriately.

The Warp Modifier rotates the mesh vertices as a function of distance around an axis. To actually be a warped surface, the axis of the Widget for the Warp Modifier needs to be at an angle to a reference plane such as a flat disk. Name: Set a name that allows you to identify this modifier easily. Enabled: When this flag is turned off, the Warp Modifier is switched off. Deg: If set, the the rotation as a function of distance in the Magnitude Graph is given in degrees per unit distance. When switched off, then it will be radians per unit distance. Magnitude: Opens the function graph to set how the rotation angle is as a function of distance from the local coordinate system set by the Widget. Widget: The Widget opens the Widget Dialog. It allows you to change the direction of the Warp Modifier. The turquoise arrow indicates the direction around which the rotation will be performed. In the example it was rotated around the original x-axis by 30 degrees. Modifiers: Warp

Math module of the Shape software. Math Module Overview The Math module is a tool to centralize parameter values that are used in more than one place in Shape. In a way similar to a spreadsheet it allows to compute variables from other variables that have already been defined further up. This makes it possible to set up a complex mathematical model of physical processes that can then be harnessed throughout the rest of Shape. Global Variables: Since the variables defined in the Math Module can be used throughout Shape, they are called Global Variables. Whenever you wish to use variables from the Math Module in other modules, make sure to enable the flag "Use global variables" where applicable. Workflow The basic workflow consists first in adding new slots for new variables in the order of dependency, if any. Then provide a meaningful name for and defining their relationships and values. They can then be used in a variety of contexts throughout Shape. Variable Names: While Shape automatically assigns a letter as a name for new variables, it is very highly recommended to change them with meaningful and descriptive names, such that they can easily be understood wherever they might appear in Shape. The name can be changed by double-clicking on the name field of the variable. Column Functionalities Variable: This column contains the name of the global variables. In order to use a variable in some other module of Shape, the name needs to match. Upper and lower case letters are not distinguished. When a new variable is added, it receives a default name in alphabetical order. Change the variable names to something descriptive of the meaning of its content. Expression: Variables get assign a value through a mathematical expression in this column. In most cases this will simply be a value. A value can be given in integer or floating point format as well as scientific format such as 1.09435e-7 for numbers that are much smaller than 1, or 3.2E15 those that are much larger than 1. In addition to numbers the field can contain more complex mathematical expression, including those combining one or more global variables that are further up in list. To avoid recursive dependencies, variables further down can not be included. Valid mathematical expression may include the basic symbolic operators +-*/ and ^, but also common textual operators which include the following reserved functions and variable names: POWER, SIN, COS, TAN, COT, RAND, CSC, ARCSIN, ARCCOS, ARCTAN, EXP, LN, LOG10, LOG2, ABS, SQRT, ROUND, ARCTANH, UNARYMINUS, LT, GT, and the irrational numbers "e" and "pi"; As mentioned above, these functions can also be written with lower-case letters. Menu bar The buttons on the top menu bar of the Math Module control the overall content of the Math Module. Calculate : The Calculate button executes all calculations that may in standby, e.g. for iterative computations. Direct calculations of variables that depend on variables further up are executed on confirming the variable name with the Enter key. Variable : The Variable button adds a new variable to the end of the list. The variables sequentially get names of single letters in alphabetical order. As mentioned above, it is recommended to change these names to something descriptive of its meaning. The variable name is changed by double-clicking on the text field. Separator : Separators are used to keep different sections of the list of variables clearly distinct. They can also function as headings if you add some descriptive text to their text fields. The separators are colored in dark blue. Constants : This button adds a set of variables that contain the most important natural constants in SI units. If you only need a subset of them, simply select the unwanted ones and delete them with the Remove button. Remove : To remove variables from the Math Module, select them and click the Remove button. You can select several variable together with Shift-Click and remove them in one operation. A confirmation dialog helps to make sure that you do not delete variables by accident.. Up & Down : To move a variable in position in the list, select the variable and click on the Up or Down button to move it by one position. Repeat the operation until the variable is in the desired position. Defaults : Use these buttons to restore the default values that variables may have in the Default field. Note that this button restores the defaults of ALL variables at once . Individual or a subset of defaults have to be reset manually. Save & Load : In order to keep sets of variable interchangeable between different projects that have similar setups, one can save the content of the Math Module in a file with the Save button. To open a saved set of variables use the Load button.

Filamentary texture generator in Shape Key Sub-S ystem: Textures Many astronomical objects, especially nebulas have filaments with random structures. This can be simulated with procedural 3-D texture s. In Shape procedural textures can be applied to physical quantities such as density, temperature, velocity, etc . These textures are multiplied on top of the spatial variation given by the Magnitude of a quantity. Open the Texture Parameter Panel by clicking on the Edit button beside the Texture label in the parameter panel of a selected modifier. Getting the right texture may require quite a bit of experimentation, often combining several basic textures. Texture Parameter Panel The Texture Parameter Panel has several sections. At the top-left is the preview window, where a single slice from the x-y coordinate plane of the 3-D texture is shown. A list of combined basic textures is below the preview window and various types of parameters are at the top-right. Transformation modifiers can be added at the bottom-right. Basic Workflow Initially the texture editor is blank. A new texture is added from a list of different types after clicking on the Add button to the right of the Textures List. The most commonly used type is the Space Convolution noise. Now a preview of the texture is generated using the default parameters. To get the texture that is needed change the parameters until a suitable result is obtained. More than one texture may be combined by adding further textures to the list. Note that the combination is done in the form of a multiplication of the texture values in the range (0-1). Therefore, the more textures you combine, locally the result becomes smaller and smaller. In the modifiers area, rotation and translation modifiers can be applied that are similar to the corresponding image modifiers applied in the Render Module. This is not descussed in more detail in this section. Textures Panel Add: Opens a dialog with a list of different types of procedural noise from which to choose. When you click on OK, the new texture is included in the list of already chosen textures. Del: Deletes the selected texture from the list. Up & Down : Move the selected texture up or down in the list. Copy & Paste: Copy stores the selected texture in a buffer. Paste pastes the copied texture as a new texture to the list. Note: As mentioned above, if there is more than one texture they are combined as a local product of their values, which range in the interval (1,0). Moving the textures up and down in the list does not change the result since multiplication is a commutative operation. However, if you explicitly name the textures, the sequence may help at keeping order General: Name: Set a name for this texture Enable: Enable or disable this texture Seamless : This parameter works together with the Distortion (see below). If a Distortion is applied that stretches the texture along the angle in a cylindrical coordinate system, a discontinuity appears at the 0 to 360 degrees transition. To prevent this enable the Seamless flag. An attempt is then made to generate a seamless texture by copying and rotating the same texture by 180 degrees and overlapping the two with a linear transition between the two that excludes the seam region. Currently, the result is a seamless texture that has a 180 degrees point symmetry. Bias: Sets a minimum intensity for the texture. If b is the bias level, now the range for the texture is (b,1). Properties: The detailed parameters for different types of textures vary. Here we discuss the example of Sparse Convolution Noise, which is the most suitable for most filamentary features in diverse nebular objects. Many of the parameters are common to all textures, others will differ. But a bit of experimentation will clarify the meaning of the differing parameters. Sparse Convolution Noise: Scale: The scale of the texture can be set separately in the three coordinate directions. By default they are looked together, i.e. when you change one of them, the other two automatically get the same value. They can be unlocked by unchecking the Lock flag. Then the values can be set independently. To asses the size of the features in the context of the model domain note that the preview window has the same size as the scene size in the Render Module. X Y Z Offset: These parameters move the texture along the corresponding axis. The units are those of the Render Module. Exponent : Controls the contrast between the highest and lowest levels of brightness. High values deemphasize initially lower values. Freq: The levels of spatial frequencies to be included in the random noise generation. Higher values will include smaller features. Type 2: This type generates a different look and overall smaller structures. Invert: inverts the greyscale, the interval (0,1) is linearly mapped to (1,0). Image Size: Sets the pixel size of the texture preview image. Z slice : Selects the slice to be shown in the preview. Changing the value moves the preview through the cube of slices along the line of sight. Distortion: This button opens a dialog that controls the re-mapping of the noise as a function of position. Analytical expressions can be set up to remap the noise in different coordinate systems. This allows the user to stretch the noise pattern in radial or circular directions. There is an example for such a distorted texture on the right. The second image is the same texture after applying the Seamless flag (see above). Below is the dialog that opens when you click on the Distortion button. It is similar to other Graphs . Make sure to set the correct coordinate system for the distortion to be applied. The example uses the Cylindrical Coordinate system and distorts the radial and the angular directions. Some experimentation with the analytic expression or point graph is likely needed to get the desired result.

Modules In this section we give an overview of the functionality of the different modules and provide links to more detailed information on how to use them and their subsystems. Click on the Module Icon to the left of the description for more information and access to video tutorials on the module. 3-D Module In the 3-D Module the geometric and most other properties of a model are set up interactively. Description Render Module This module takes care of the rendering of image and position-velocity diagrams and a number of settings for other render options. Description Physics Module Radiation transport properties such as emissivity, absorption or scattering are set up as materials (species) in the Physics Module. Description Desktop Module The Desktop Module is your hub to all the other modules, project files, ShapeX configuration and more. Description Video Tutorial Math Module The Math Module allows you to set up variables and relations between them that can then be used throughout Shape as "global variables". Description Modifier Module The Modifier Module lists all modifier that are currently in use and allows you to change parameters of a selection of modifiers simultaneously. Description Maps Module The Maps Module displays channels maps of the 3-D model. The number and velocity range between the first and last channel can be set up. Description Animation Module Most parameters in Shape can be animated over time. This can be used to generate time variation of the models either for scientific modeling of time varying phenomena or for visualization purposes. Description Filter Module Filters for various physical quantities can be defined here. They can then be applied to objects in the 3-D Module. Description Movie Module In the Movie Module one or more animation sequences can be concatenated to a movie and exported for viewing with an external movie player. Description Export Module The Export Module exports the 3-D model into various output formats that can then be used as data for external use. Description Hydrodynamics Module Shape is the first astrophysical tool to introduce an interactive mesh-based setup for such simulations without the need of programming or scripting by the user. Description

Key Sub-S ystem: Coordinate Systems There are several coordinate systems defined in Shape. First of all in most contexts the coordinates may be defined in Cartesian, Spherical or Cylindrical coordinates. Note that for the spherical coordinates the label convention is that of the North America, with q being longitude and f the latitude. Hierarchy of coordinate systems: In addition to the types of coordinate system, there is a hierarchy that determines the origin and orientation of the coordinates. First there is the global "world" coordinate system that is fixed and everything else is embedded in this system. The orientation of the world coordinate system is show by the colored coordinate axes in the lower left corner of the 3D views in the 3D Module (see images on the right). Note that it is not centered on the center of the coordinate system and only provides a visual cue of the orientation. The colors of the xyz axes follow the common order of the color channels rgb (red, green, blue), respectively. Every object has its own "local" coordinate system , that may move around in the world coordinate system depending on the types of modifier that are applied. As an illustration compare the two images on the right. In the first one the spherical mesh and the density distribution are centered on the world coordinate system. No changes have been made to any positions. In the second example a Translation Modifier has been applied. It moves the mesh away from the World Origin. Not only the mesh is moved but the density distribution goes along. Similarly, the Rotation Modifier will rotate the density structures along with the mesh, since thee local coordinate system changes . The translation, rotation and scale operations can be interactively handled using the corresponding Move, Rotate and Size tools in the System tab that is located to the left of teh 3-D views. It is important to note that, contrary to the Translation and Rotation modifiers, the Size tool and Modifier does NOT change the scaling of the local coordinate system. This would cause too many practical problem during modeling. It only changes the mesh. This is similar to the Displacement modifier which only moves the mesh, not the coordinate system. For rotating only the mesh, operators such as the Twist modifier can be applied. In third place there is the "widget" coordinate system that is applied to some modifiers. They often need to be centered at different positions within an object, which can be achieved by moving and rotating the coordinate system of the modifier using the Widget tool or Widget dialog . Different modifiers have independent widget coordinate systems. By default modifiers follow the local coordinate system , meaning that their widget coordinates are coincident with the local system until the widgets themselves are changed. But the coordinate center that a modifier refers to can be changed by changing the widget either interactively with the Widget tool or numerically by opening the Widget dialog from within the modifier panel. To open the Widget dialog click on the Widget Edit button at the bottom of the modifier panel. In the image on the right under the widget dialog the object mesh remains at the world origin, while the density distribution is off-center at the position of the widget . The widget itself is represented by arrows. Each modifier may have independent widget, i.e. local coordinate system positions. In the following image shows the application of the Displacement modifier , which moves only the mesh , not the coordinate system. The mesh container changed position, but the density distribution remained centered on the world coordinates. An additional fourth coordinate system is that of the observer or camera . These are the coordinates seen in the "shape view port" of the 3D Module and the rendered image in the Rendering Module . OPERATOR ORDER MATTERS! In the modifier stack several rotations, translations and other operators can be applied one after the other. The result of such combined operations, in general, strongly depends on the order in which they are executed. Therefore the order of the operator in the modifier stack is very important. For instance, translation can be combined in any order (they are commutative ), rotations among themselves and rotations together with translations can not. Coordinate Display Options Dialog: The Options Dialog that opens by clicking on the wrench icon on the menu bar of the 3D Module allows you to customize the display of a Cartesian coordinate mesh within the 3D views. It can helps as a reference during the modeling process. An example is show in the bottom image in the column on the right. A little experimentation should clarify the meaning and effect of the various parameters.

The Squish Modifier changes the distance of a vertex perpendicular to a plane (default: local xz-plane), The action is similar to the Squeeze Modifier , except that it´s planar, not radial around an axis. The Magnitude dialog allows you to define the squish amount as an Analytic Function of position along the reference axis. You can also use a Point graph where you can generate an arbitrary function by manually placing points and setting the spline interpolation. To do this, select Point from the Function drop-down list under the graph. The example graph on the right shows the way it was done for the example mesh displayed below. Widget: The Widget opens the Widget Dialog. It allows you to change the direction of the Squish Modifier. The purple arrow will indicate the direction of its action. Modifiers: Squish

Filamentary texture generator in Shape LEGAL AND PRIVACY INFORMATION Impressum Dr. Wolfgang Steffen Contact: e-mail: contact@ilumbra.com Responsible for the content: Dr. Wolfgang Steffen Hautzenbergstrasse 1 67661 Kaiserslautern Germany Copyright 2021 Owners: Dr. Nico Koning (ilumbra), Dr. Wolfgang Steffen (ilumbra) Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal use the Software without restriction, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. 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