Reference: General Topics

Transcription

1 Reference: General Topics Topics Model-level Settings Defining Objects Dependency Examples How Much Accuracy Is Enough? Fairness of Curves and Surfaces Relabeling Curves and Snakes Making Hardcopy Pictures of MultiSurf Models The MultiSurf Model File MultiSurf 4.0 Reference: General Topics 4-1

2 Model-level Settings Topics Model-level Settings Topics Symmetry Images Units Home View Places Divisions Multiplier Update Extents Model Comments Symmetry Images Set in File/ New or Settings/ Model. If your entire model has mirror symmetry across one or more of the coordinate axes, or rotational symmetry around one of the coordinate axes, you can make use of MultiSurf s symmetry images. This means that you can create points, curves, and surfaces that represent a portion of the model (we call this the real part of the model), then turn on the symmetry images to see the entire model. View/ Symmetry Images (or <F5>) is a toggle, allowing you to display the model with all images showing or with only the real objects visible. Note 1: If your model has only partial symmetry, or symmetry about an axis other than one of the coordinate axes, use mirror and rotated objects (MirrSurfs, etc.) rather than symmetry images. Note 2: If you will be using the weight schedule calculations, symmetry images may affect the weight values you assign to some objects. See Program Operation: The Menus - Calculations/ Weight Schedule. Mirror Symmetry Mirror symmetry is symmetry across one or more of the coordinate-axis planes. For example, most boat designs have complete (model-level) symmetry with respect to the Y=0 plane. In this case, it is generally advantageous to use Y-symmetry so you only define objects on one side of the Y=0 plane and the mirror image objects are implied by mirror symmetry. Or, suppose you design a shampoo bottle that is symmetric 4-2 Model-level Settings December 1998

3 Model-level Settings Units with respect to both the X=0 and Y=0 planes. In this case, it could be advantageous to use both X and Y symmetry; then the real objects would comprise only one quarter of the model. Rotational Symmetry Rotational symmetry is symmetry around one of the coordinate axes. In the rotational symmetry, n is the total number of copies to be displayed, including the original. n-fold rotational symmetry is present if the model looks the same after being rotated 1/n revolution (360/n degrees) about the indicated axis. For example, a 5-bladed propeller would have 5-fold symmetry about the shaft axis. In this case, you could design only one blade (and 1/5 of the shaft, hub, etc.); turning on the symmetry images would display the other 4 blades, which would be completely identical to the original (look at model file PROP3.MS2). Mirror/rotational combinations Rotational symmetry can be combined with mirror symmetry, but only to this extent: a model can have mirror symmetry with respect to the same axis as its rotational axis. For example, if rotational symmetry with respect to the Z-axis is specified, this can be combined with mirror symmetry with respect to Z. If you do not specify symmetry, the model will have no symmetry images. Units Home View Set in File/ New or Settings/ Model. The units setting is a property of the entire model. You can choose any of the following: Symbol Length unit Weight unit None (no units specified) (no units specified) ft-lb foot pound m-kg meter kilogram ft-st foot short ton (2000 lbs) ft-lt foot long ton (2240 lbs) m-mt meter metric tonnes (1000 kg) in-lb inch pound If you choose None, one unit in the model can stand for any length unit you want feet, meters, cubits, miles, or millimeters. Note: You can simultaneously open models with different units. Set in File/ New or Settings/ Model. Home view is the view in which MultiSurf first displays a model on the screen. It is also the view you can return to at any time by pressing the <Home> key. The Home view can be perspective (default) or orthographic. If you turn on the orthographic checkbox, the view will be orthographic rather than perspective. If you MultiSurf 4.0 Model-level Settings 4-3

4 Model-level Settings Places do not specify Latitude, Longitude for this view, the program will use the default Home view: Latitude -30, Longitude 60, perspective. Places Set in File/ New or Settings/ Model. Set the number of decimal places displayed for linear dimensions. The default is 3 decimal places. Divisions Multiplier Set in File/ New or Settings/ Model. The divisions multiplier is the factor (1-10) MultiSurf uses for multiplying the subdivisions of all curves, snakes, and surfaces (except TabCurves and TabSurfs): When the divisions multiplier = 1 (the default), the divisions used in the calculations are those specified for each object. When the divisions multiplier is 2-10, MultiSurf multiplies each object s subdivisions by the divisions multiplier factor when it makes calculations. This tool allows you to work with low divisions for speed, then to crank the divisions up uniformly when you need more accurate construction output (for construction drawings or N/C cutting), avoiding the cost of Accurate mode calculation. For more details, see Reference topic - How Much Accuracy Is Enough? on page 16. Update Extents Accessed in File/ New or Settings/ Model. MultiSurf frames the image of your model on the screen by calculating the extents of a bounding box. If you find that your model is not being framed nicely on the screen, most likely the extents of the bounding box have gotten out of date. To fix this: go to Settings/ Model and click the <Update Extents> button. Model Comments Set in File/ New or Settings/ Model. Model-level comments allow you to include some information in your file that will help you identify the model, for example: reindeer model 7 for Santa Claus 8/3/94 (note: please deliver in time for training 11/6/94) or Monson 30 lobster boat - modification 4 <Enter> creates new line. 4-4 Model-level Settings December 1998

5 Defining Objects Topics Defining Objects Topics Overview Overview Summary of Attribute Data Attributes Required by All (or Almost All) Entities Additional Curve and Snake Attributes Additional Surface Attributes The Variable Portion of an Object Definition We use the term entity to refer to a kind of geometric object. A particular instance of that entity is called an object. To create an object, you: (1) select the kind of entity it is to be (2) specify the particular data for that individual object, according to the attribute data required for that kind of entity For instance, if you want to create an AbsPoint (Absolute Point), you will need to give the following attribute data (or accept the defaults): object name color visibility X, Y, and Z coordinates In this chapter, we explain the attributes common to most entities or entity classes (points, curves, etc.). Summary of Attribute Data All kinds of entities require at least the following attribute data: object name color visibility MultiSurf 4.0 Defining Objects 4-5

6 Defining Objects Attributes Required by All (or Almost All) Entities Note of exception: BGraph, KnotList, ObjectList, and Relabel entities do not have color or visibility attributes. All kinds of entities may have the following optional data: layer and weight user data All curves and snakes require the following data: t-divisions t-subdivisions relabel All surfaces require the following data: u-divisions u-subdivisions v-divisions v-subdivisions orientation All solids require the following data: u-divisions u-subdivisions v-divisions v-subdivisions w-divisions w-subdivisions orientation Beyond these data categories, the balance of each object definition is the particular data required to specify an object of that entity type. This data may include coordinate values, type, names of other objects ( supports ) in the model file, and/or graph functions that control the distribution of various parameters. In addition, you may specify optional layer and weight data and/or optional user data (object comment). Attributes Required by All (or Almost All) Entities Object name ALL objects have a name. An object name labels a specific object. For instance, top might be the name of an AbsPoint (absolute point) located at x=0, y=0, z=10. Object names may be used in the creation of subsequent objects. Object names are userdefined and can be up to 16 characters. Duplicate object names are not allowed within a model. You can use any combination of letters and numbers, but most punctuation marks and symbols are illegal. The following are not allowed in object names: space, semicolon (;), colon (:), comma (,), period (.), parentheses ( ), slash (/), asterisk (*), question mark (?), double quote mark ( ), braces ( { } ), plus sign (+), minus sign or hyphen (-), caret (^) The underscore character _ can conveniently be used where you would like a space or a hyphen; for example, left_side. 4-6 Defining Objects December 1998

7 Defining Objects Attributes Required by All (or Almost All) Entities Object names ARE case sensitive. If you so chose, you could use ABC, abc, and AbC to name three different objects. Under no conditions would the program recognize these three names as referring to the same object. Sometimes we like to use names that tell what physical object the geometric object represents; sometimes we prefer, especially for points, short names in a simple sequence. When speaking of objects, we often use the entity name and object name together to identify an object, for example AbsPoint P3 or CCurve sheerline. The entity name (e.g. AbsPoint or CCurve) is like a family name, and the object name (e.g. P3 or sheerline ) is like a given name. User data An optional string of user data can be attached to an object. It can be a remark associated with the object or it can convey user data that is meaningless to MultiSurf. 40 characters maximum embedded spaces are okay; embedded quotes are not allowed. Layer and weight Layer and weight data can be attached to specific objects: layer is an integer (0-255); default = the current layer unit weight can be a positive or negative value; decimals okay; default = 0 The definition of unit weight varies by entity class: for points, it is weight for curves, snakes, and contours, it is weight/unit arc length for surfaces, it is weight/unit area for solids, it is weight/unit volume Note that layer and weight have no meaning for Relabels, KnotLists, ObjectLists, and BGraphs. Also, weight has no meaning for planes, frames, or WireFrames. Color Almost all entities have a color attribute (BGraph, KnotList, ObjectList, and Relabel do not). color is an integer, 0 or greater. The distinct colors available run from 0 to 15 inclusive (see table below). 0 black 8 gray 1 blue 9 bright blue 2 green 10 bright green 3 cyan 11 bright cyan 4 red 12 bright red 5 magenta 13 bright magenta 6 brown 14 yellow 7 white 15 bright white To accommodate those of you who transfer files to programs such as AutoCAD which use a color range, you can specify a color code for colors over 15. Here s how MultiSurf interprets color codes: it divides the code number by 16 and uses the remainder as the color number for display. For example, color 42 will appear in MultiSurf the same as color 10 (42 16 = 2 with a remainder of 10). MultiSurf 4.0 Defining Objects 4-7

8 Defining Objects Additional Curve and Snake Attributes Note that, depending on your computer system and the brightness of your screen, the colors may appear somewhat differently (e.g. red may appear pink or brown). Visibility Almost all entities have a visibility attribute (BGraph, KnotList, ObjectList, and Relabel do not). Some visibility options vary according to the class of entity; the option all share is that any entity can be hidden (or not hidden, of course!). The visibility options can be used singly or in combination. Additional visibility options For curves and snakes, you can display: the curve tickmarks along the curve/snake at intervals of 0.1 in the parameter t the polyline connecting the control points (for BCurves and CCurves) For surfaces, you can display: u=constant lines v=constant lines boundary lines net (for NURBSurf, BSurf) For solids, you can display: u-direction lines v-direction lines w-direction lines boundary lines Note: The solid display options are not parallel to those for surfaces. Additional Curve and Snake Attributes divisions: t-divisions and t-subdivisions All curves and snakes have t-divisions and t-subdivisions attributes. The number of t-divisions x t-subdivisions specifies the number of line segments the program will use to approximate a curve (or snake) for display and internal tabulation. t-divisions and t-subdivisions are both integers (1-255). The greater the total number of divisions (divisions x subdivisions), the more closely the curve screen display will approximate the curve itself (which is composed of an infinite number of points). Let s look at the displays of two B-spline curves whose definitions differ only in the number of divisions. The first, specified with divisions = 5x1, is displayed with 5 line segments (Fig. 1, left); the second, specified with divisions = 40x1, is displayed with 40 line segments (Fig. 1, right). Note that divisions = 5x8 would look exactly the same as 40x Defining Objects December 1998

9 Defining Objects Additional Surface Attributes divisions = 5x1 divisions = 40x1 Fig. 1. A curve drawn with 5x1 t-divisions (left) and 40x1 t-divisions (right). The two main reasons you might want to use more than 1 t-subdivision are: To coordinate divisions between a surface and its supporting curves. The easiest way to coordinate the divisions is to give the curve and the u-divisions of the surface the same number of divisions say 10x3. For details, see Additional Surface Attributes: divisions - Coordinating divisions on page 10. To have more than 255 divisions. For instance, if your object is a 10-turn helix (angle = 3600 degrees), 255x1 divisions might not be enough for adequate accuracy since the calculations use the polyline in place of the actual curve each turn would be represented by only 25.5 segments. For details about the parameter t, see Concepts - Parametric Curves. relabel All curves and snakes have a relabel attribute, used to distribute the parameter t along the curve (or snake). To have MultiSurf use its default labeling, which produces the natural labeling for the curve, you just use an asterisk * for relabel in the curve s definition. In most instances, the default labeling is quite satisfactory. If you choose to relabel a curve, you must create a Relabel object that specifies the new labeling and include the Relabel object s name in the curve s definition. Note that a Relabel object CANNOT be used to reverse the t-orientation of a curve depending on the kind of curve, you would do that by reordering its control points (e.g. BCurve), changing its type (e.g. EdgeSnake), etc; or you might create a SubCurve/Snake or a BSubCurve. For details on relabeling curves, see Reference section Relabeling Curves and Snakes on page 30. Additional Surface Attributes divisions: u and v divisions and subdivisions All surfaces have u-divisions, u-subdivisions, v-divisions, and v-subdivisions attributes. The number of u (or v) divisions x subdivisions = the number of line segments the program uses to represent the surface in the u (or v) direction for display and internal tabulation. u-divisions, u-subdivisions, v-divisions, and v-subdivisions are all integers (1-255). The greater the total number of u and/or v divisions, the more closely the surface screen display will approximate the surface itself (which is composed of an infinite number of points). MultiSurf 4.0 Defining Objects 4-9

10 Defining Objects Additional Surface Attributes In addition, the number of u-divisions (or v-divisions) establishes the spacing of the u=constant (or v=constant) lines drawn in a wireframe display. For instance, if a surface is defined with 10x3 20x1 divisions, MultiSurf will tabulate the surface internally with 30 segments in the u-direction and 20 segments in the v-direction. In the display, if the u=constant lines are visible, there will be 11 of them (0,.1, , 1.0; Fig. 2, top). If the v=constant lines are visible, there will be 21 of them (0,.05,.10,....95, 1.0; Fig. 2, bottom). For details about the parameters u and v, see Concepts - Parametric Surfaces. Z u-constant lines Y v 0,0 u X v-constant lines v 0,0 Y Z u X Fig. 2. u=constant and v=constant lines on a surface. Coordinating divisions It is very helpful to get into the habit of coordinating the divisions of surfaces with the divisions of their supporting curves. In general, coordination of divisions produces a more accurate stored model, and it is necessary for the display of views such as Surface Curvatures. Here s what we mean: when creating a lofted surface, use the same number of divisions on each of the master curves, and also make sure that the master curve t-divisions x t-subdivisions is the same as, or a multiple of, the surface u-divisions x u-subdivisions. E.g. you might use: u divisions = 8x4 and t divisions = 8x4 (surface and curve divisions are the same) or u divisions = 4x1 and t divisions = 8x4 (curve divisions are a multiple of surface divisions) orientation (normal orientation switch) All surfaces have an orientation attribute. The normal orientation switch tells which way the positive normal of the surface points. While orientation has NO effect on the appearance of the surface in any MultiSurf view, there are some situations where you do need to be aware of it: 1. For offset points, curves, and surfaces, orientation of the supporting surface controls the direction of the offset Defining Objects December 1998

11 Defining Objects Additional Solid Attributes 2. If you are exporting.pat files for computational fluid dynamics (CFD) or finite element method (FEM) programs, they may be sensitive to panel inside/outside orientation. 3. If you ll be expanding ruling files (.RUL) in MSDEV or compound-curved plates in MSPLEX, you ll need to use the normal to specify in which direction from the surface the panel thickness will be point the normal in the direction of the thickness. orientation can be either 0 or 1. Probably the easiest way to figure out which value to assign is to create your surface with the default, then use the Tools/ Orientation display to show you the way the normal is pointing. If the arrow (displayed at the center of the surface) is not pointing in the direction you want, you know you need to edit the object and change the orientation value (from 0 to 1, or vice versa). Another way to figure out which value to assign the switch is look at the surface with the u-direction going to the right and the v-direction going up. If you want the normal to point toward you, then you specify orientation = 0 (Fig. 3); if you want the normal to point away from you, then you specify orientation = 1. X for normal pointing up toward you, orientation = 0 for normal pointing away from you (through the page) orientation = 1 v u Fig. 3 Figuring out the value for the normal orientation switch. Additional Solid Attributes divisions: u, v, and w divisions and subdivisions All solids have u-divisions, u-subdivisions, v-divisions, v-subdivisions, w-divisions, w- subdivisions attributes. The number of u (or v or w) divisions x subdivisions = the number of line segments the program uses to represent the surface in the u (or v or w) direction for display and internal tabulation. u-divisions, u-subdivisions, v- divisions, v-subdivisions, w-divisions, and w-subdivisions are all integers (1-255). The greater the total number of u, v, and/or w divisions, the more closely the solid screen display will approximate the solid itself (which is composed of an infinite number of points). MultiSurf 4.0 Defining Objects 4-11

12 Defining Objects The Variable Portion of an Object Definition In addition, the number of u-divisions (or v- or w-divisions) establishes the spacing of the u-constant (or v-constant or w-constant) lines drawn in a wireframe display. 0,0,0 u w v-direction lines w-direction lines v u-direction lines Fig. 4. u, v, and w-direction lines on a solid. For instance, if a solid is defined with 6x4 4x2 2x2 divisions, MultiSurf will tabulate the solid internally with 24 segments in the u-direction, 8 segments in the v- direction, and 4 segments in the w-direction (Fig. 4). Note: The visibility options for solids do not parallel those for surfaces. Solid display options show the u-direction lines (not the u= constant lines), etc. These display lines run in the same direction as the orientation marks for u, v, and w. For details about the parameters u, v, and w see Concepts - Parametric Solids. orientation Like surfaces, solids have an orientation attribute which can be either 0 (normal) or 1 (reversed). Solid orientation is different from surface orientation currently its only effect is to reverse the sign of volume and weight. The Variable Portion of an Object Definition Beyond keyword, name, color, visibility, divisions (if any), optional data (if any), relabel (if any), and normal orientation switch (if any), the balance of each object definition is the specific data required to specify an object of that entity type. For instance, all the data required to complete an AbsPoint object is its three coordinates X, Y, Z. The rest of the data for a CCurve is the value for type and the list of control points the curve goes through. The rest of the data for a CLoftSurf is the value for type and the list of master curves the surface is sprung over. In addition, you can lock an object if you don t want it to be directly editable. For details, see Program Operation: General Topics - Creating and Editing Objects - Locked/unlocked status. For details, see the particular entity in the Reference section Entity Descriptions Defining Objects December 1998

13 Dependency Examples Example 1 Dependency Examples Example 1 Let s look at two examples of relationships between objects in MultiSurf models. We ll introduce yet another way of representing those qualitative dependencies a diagram called a directed graph or digraph. The digraph in Figure 2 is for a simple model composed of five points named P1 - P5, and a spline curve named sheerline (Fig. 1). Z 'sheerline' 'P3' 'P2' Y 'P1' 'P5' 'P4' X Fig. 1 P1 P2 P3 P4 P5 sheerline Fig. 2 Each oval in the digraph represents an object, and each arrow represents a dependency. The direction of the arrow tells the direction of the dependency: sheerline depends on P5, not the other way around. The qualitative information that the spline ends at P5 is permanently retained by the program, permitting MultiSurf 4.0 Dependency Examples 4-13

14 Dependency Examples Example 2 automatic update of the spline whenever P5 is moved. The spline s dependence on the other four points is similarly retained. Likewise, subsequent objects that depend on the spline sheerline for their location or shape will be automatically updated to conform to the spline s new shape, etc., etc. throughout the entire model. Example 2 Let s explore another related example. The motivation for this is the following: the sheerline (i.e., the line where the edge of the deck joins the hull of a boat) is visually one of the most prominent lines in a boat design, and consequently it is the focus of much aesthetic consideration. Many designers are aware of a rule that states that it is highly desirable that such a visual line lie in a single plane. The sheerline created in Example 1 might lie in a plane it s certainly not far from that but there s no assurance that it actually does, and certainly no assurance that it will continue to lie in a plane as we move the points around in shaping it. Example 2 introduces this constraint to the earlier sheerline of Example 1, also adding the mirror-image sheerline for the other side of the hull. 'centerplane' 'sheer_plane' 'sheer_port' (mirror curve) Z X 'P5' 'Q4' and 'P4' (very close together) 'P3' 'P1' 'sheerline' (basis curve for 'sheer_port') 'Q2' and 'P2' (very close together) Fig. 3 The five points P1 - P5 are the same ones as in Example 1. The object named sheer_plane is a plane, made by one of several methods MultiSurf provides, in this case specification of three points ( P1, P3, and P5 ) through which the plane must pass. That plane was used to form two projected points Q2 and Q4, which are the perpendicular projections of P2 and P4 onto the plane. In this way we have collected five points that durably lie on a plane P1, P3, and P5 do so because the plane was constructed to pass through them, and Q2 and Q4 do so because they are projections onto the plane and we are prepared to utilize an important property shared by most of MultiSurf s curve entities: if all the control points lie in a plane, so does the entire curve. sheerline is the planar curve we created; it is a C-spline curve passing through P1, Q2, P3, Q4, and P5. Now we are free to shape the curve by adjusting any of the five points P1, P2, P3, P4, P5 (we move the projected points Q2 and Q4 by moving their basis points P2 and P4 ); the resulting curve will always lie in a plane. centerplane is another plane, defined by a Y-coordinate of zero another way of constructing a plane. It is used as the mirror for creating sheer_port, the mirror image of sheerline. (Note: if our entire model were going to have mirror symmetry across the centerplane, we could specify model-level Y symmetry and use symmetry images instead of a mirror curve to create the port side.) Y 4-14 Dependency Examples December 1998

15 Dependency Examples Example 2 P1 P3 P5 P2 P4 sheer_plane Q2 Q4 sheerline centerplane sheer_port Fig. 4 Now consider the series of qualitative dependencies we have formed. The digraph is a good deal more complex (Fig 4). A change in P5 now changes sheer_plane, which changes Q2 and Q4. So three of the points supporting sheerline are changed, and this changes the whole curve. Likewise the mirror image curve is all changed. But... MultiSurf takes care of all these cascading changes for you. The result is that you are free to move any or all of the points P1 to P5, adjusting the shape of sheerline to your heart s content, always assured that your sheerline lies in a plane and that the mirror image is 100% perfect. We are tempted to go on and on, using the port and starboard sheerlines as edges of a deck surface, forming a hull surface that joins the sheerlines from below, etc., etc. building a boat. But we do that in a tutorial, so we ll not repeat it here. Our point at the moment is that by storing the qualitative relationships between objects in the form of instructions for building the model (rather than the detailed present geometry of the model), MultiSurf can preserve these relationships through many levels of dependency, providing you with unprecedented freedom in creating, revising, and reusing complex geometric models. MultiSurf 4.0 Dependency Examples 4-15

16 How Much Accuracy Is Enough? Topics How Much Accuracy Is Enough? Topics Fast vs. Accurate Divisions Coordinating divisions Number of divisions Fast vs. Accurate As we and other MultiSurf users have been capitalizing on the capability of the Windows version to create highly complex models, we have collided head on with the speed limitations of Accurate calculation mode it is SO SLOW! Although we used to recommend use of Accurate calculations when making construction drawings or N/C cutting files, we now find that in our office we rarely use Accurate mode any more, because we can get construction quality accuracy in Fast mode by assigning enough divisions and coordinating the divisions between surfaces and their supporting curves. In Fast mode, many calculations are performed quickly by linear interpolation in lookup tables. In Accurate mode, all the direct calculations implied in the model are actually performed. This often can take hours instead of the seconds it takes in Fast mode, but the results should be accurate to the full precision displayed. In Fast mode, by increasing divisions, you can approach Accurate mode accuracy, and in some cases actually attain it. In fact, though, you don t need to get there there is enough inaccuracy in the physical construction process that Accurate mode accuracy is rarely (if ever) achieved outside of MultiSurf. So how much MultiSurf accuracy is enough? Divisions There are two factors involved in assigning appropriately accurate divisions: matching or coordinating the divisions between surfaces and their supporting curves choosing the number of divisions x subdivisions for each surface 4-16 How Much Accuracy Is Enough? December 1998

17 How Much Accuracy Is Enough? Divisions Coordinating divisions The easiest way to coordinate divisions between surfaces and their supporting curves is to match their divisions x subdivisions (see discussions in Coordinating divisions on page 10 and in Coordinating divisions: an example on page 27). You can accomplish this automatically if you use the default divisions 8x4 when creating curves, and the default divisions 8x4 8x4 for a surface built off those curves. When divisions are matched, there is no tabular discrepancy between a surface and its supporting curves and the result is exactly the same as accurate mode. Number of divisions Increasing the number of divisions improves the accuracy of output in proportion to the square of divisions x subdivisions, e.g. doubling divisions x subdivisions reduces maximum error by a factor of four. In practical terms, we typically start with the MultiSurf default divisions x subdivisions = 8x4 (32) for display and working drawings. Occasionally, where there is tight curvature, we go higher. Then, for full-scale lofting and NC cutting, we generally double those default divisions using the global Divisions Multiplier (Settings/ Model/ Divs. Multiplier) when this value is set to 2, the program doubles the subdivisions of all curves and surfaces when they are used in calculations. The table below shows 4 examples you can use as guidelines in choosing divisions for your models. DEMO.MS2 Huckins 69 Model Direction of Divisions Display and Working Drawings Full-scale Lofting and NC Cutting Comments DEMO.MS2 transverse longitudinal top panel: trans. top panel: long. bottom: trans. bottom: long. transverse longitudinal 10x3 10x3 10x5 10x5 Huckins 69 8x4 8x4 10x1 8x4 8x8 8x8 10x2 8x8 match the longitudinal divisions of both panels Stanley 21 8x4 8x4 8x8 8x12 longitudinal divisions higher in order to get a smooth rabbet line Chocks for Hinckley Picnic Boat yellow inside red top green side 8x4 8x4 8x2 8x4 8x2 4x1 8x8 8x10 8x4 8x10 8x4 4x1 adjoining edges of these surfaces have same divisions MultiSurf 4.0 How Much Accuracy Is Enough? 4-17

18 How Much Accuracy Is Enough? Divisions Stanley 21 Hinckley Picnic Boat chock As you can see, there is no absolute rule. You ll need to do a bit of experimenting, but there is simply no point to using 50x4 divisions when even 16x4 would be overkill. One way to check in on your choice is to save a 2D file and look at it in your CAD program zoomed in at 1:1 you can see the actual straight line segments How Much Accuracy Is Enough? December 1998

19 Fairness of Curves and Surfaces Topics Fairness of Curves and Surfaces Topics Fairness of Curves Curvature profiles: basic reading Fairing the sheerline Fairing the profile line Fairness of Surfaces Surface curvatures Some properties and applications of surface curvatures Examples Fairness of Curves Boats of almost all types are aesthetic objects. Sweet or fair lines are widely appreciated and add great value to many boats at very low cost to the designer and builder. Especially when there is no conflict with performance objectives, it is almost criminal to design an ugly line when a pretty one would serve as well. What constitutes a fair line? Simplicity is one simply stated criterion. A curve should be no more complex than it needs to be to serve its function. It should be free of unnecessary inflection points (reversals of curvature), rapid turns (local high curvature), flat spots (local low curvature), or abrupt changes of curvature (such as a straight line joining a circular arc). All these qualities describe fairness in terms of the distribution of curvature along a curve. Consequently a tool for visualizing the distribution of curvature is a powerful aid in shaping pleasing curves. This is Tools/ Curvature Profile. As an example of its uses, we ll take you through removing some imperfections from a particular sample model CLFT7X4.MS2 (Fig. 1). Go ahead and open CLFT7X4.MS2 (it s in \MSURFWIN\SAMPLES). This is a C-lofted hull with 7 BCurve master curves. Note that an important method for designing fair hulls is to limit the number of master curves and control points. When there are only 3 or 4 master curves, the longitudinal lines practically have to be fair they are just splines held at a very few points. When you increase the number of master curves or control points, you increase your control over the shape of the surface, but you also greatly increase the potential for unfairness and the need to test for fairness, and to fiddle with the control points until you achieve it. MultiSurf 4.0 Fairness of Curves and Surfaces 4-19

20 Fairness of Curves and Surfaces Fairness of Curves 'sheer' 'profile' Fig. 1 CLFT7X4 has 7 master curves supporting its surface. We have added two more curves than we include in most C-lofted hull models (Fig. 1): sheer a type-3 CCurve through the first (top) control point of each master curve profile a type-3 CCurve through the last (bottom) control point of each master curve First we ll look at the sheer curve, and its control points. To clear out the picture, turn on Curve Nametags, select sheer and profile, Select/ Supports/ 1st Generation, Select/ Invert Selection Set, Hide Selection Set. Now, go to the <z> plan view (Fig. 2a). The t=0 end of the curve is at the bow (right end). This projection of the curve appears fair just looking at the screen image. Let s see if the curvature profile (Fig. 2b) gives us any additional information select sheer, Tools/ Curvature Profile, then Zoom to Selection. Fig. 2a. Lat 90. Curvature profiles: basic reading Fig. 2b. Lat 90. First, the graph is a plot of the curvature of the selected curve vs. arc length in its current screen projection. For instance, here s how the graph changes as we move to Lat 60 (down arrow 3 times; Fig. 3) and Lat 30 (down arrow 3 more times; Fig. 4). Fig. 3. Lat 60. Fig. 4. Lat 30. The circle around one end of the graph axis shows which end of the graph corresponds to the t = 0 end of the selected curve. Curvature is the inverse of the 4-20 Fairness of Curves and Surfaces December 1998

21 Fairness of Curves and Surfaces Fairness of Curves radius of curvature (a tight curve has high curvature, a straight portion has zero curvature). The sign is significant: positive curvature indicates the curve is curving toward the left as you move in the direction of increasing t negative curvature indicates the curve is curving toward the right as you move in the direction of increasing t A zero crossing of the horizontal axis in the curvature profile is an inflection point on the curve. Putting yourself in bug-mode may be the easiest way to understand this. How to be a bug? Just imagine it you are your favorite bug, walking on your screen in the direction of increasing t, along the selected curve. If as you walk you are curving to your right, that portion of the curve has negative curvature. And conversely, if as you walk you are curving to the left, that portion of the curve has positive curvature. Fairing the sheerline in plan view Let s go back to the original <z> plan view (Fig. 5a): Fig. 5a. Lat 90. The graph for sheer in this view (Fig. 2) has a shallow W shape, and all the curvature values are negative. So what does this tell us? Well, it tells us that if we were the proverbial bug walking along the screen on top of sheer, we would everywhere be curving toward the right, i.e. upward toward the top of the screen. This much is obvious by looking at the sheer on the screen. Not obvious looking at the screen image is that there is a relatively flat spot near the middle of the curve, indicated in the curvature profile by the point in the middle of the W, where the curvature is only about half as great as it is a few feet forward and aft (remember that we re looking at negative curvature right now). The visible corners in the curvature plot are at the knots of the spline curve. For a type-3 CCurve, the knots are at each of the control points, except for the ends and the first control point inside each end. The flat spot is therefore associated with the middle point P41 (Fig. 5b). Because the curvature is not as negative as it should be, this point has to be eased downward on the screen, i.e., toward positive Y. 'P11' 'P71' 'P61' 'P51' 'P41' 'P31' 'P21' Fig. 5b Here s what happens when we move P41 just.05 units outboard. We could do this with Edit/ Move Point and type in Y = 4.59, but we can see the subtle changes better by nudging the points with <Ctrl>+the arrow keys (and <Shift>+Ctrl>+arrows, which nudges by 1/10 of the nudge increment). Go to Settings, Dragging and set Nudge to.01 (we ll be nudging by.01 and.001 in this demonstration). Select P41 ; hold down <Ctrl> and press the down arrow key 5 times, <Enter> to accept (Fig. 6). We d be hard pressed to notice the difference MultiSurf 4.0 Fairness of Curves and Surfaces 4-21

22 Fairness of Curves and Surfaces Fairness of Curves in the screen image of the curve itself, but the new curvature profile shows the flat spot is gone. Now the curvature is nearly constant (approximately a circular arc) from P31 to P51, tapering off moderately toward both ends of the boat. We suggest this is an improvement. Fairing the sheerline in profile view Fig. 6. Lat 90 (edited sheer). Now let s look at the sheerline in profile, the <y> view (Lat 0, Lon 90; Fig. 7a). Here the curvature profile is an M shape (Fig. 7b). There are two relatively flat spots, indicated by the peaks of the M ; they are at P31 and P51. In fact, at P31 the curvature is positive, showing this part of the curve is actually curving downwards (your bug would turn toward the left for a little bit). The zero crossings of the curvature profile are two inflection points. 'P71' 'P61' 'P51' 'P41' 'P31' 'P21' 'P11' Fig. 7a Fig. 7b. Lat 0. This curvature distribution could be smoothed out by raising P41, but let s assume that for some reason we want to hold this point where it is. Then we can lower P51 and P31 instead. Let s see what happens. Select P51 ; <Ctrl>+ down arrow twice (Z = 1.99). Select P31 ; <Ctrl>+down arrow twice; then <Shift>+ <Ctrl>+down arrow 7 more times, <Enter> (Z = 2.34; Fig. 8). Fig. 8. Lat 0. Again, the results in the screen display of the curve itself are barely, if at all, noticeable; but now the curvature profile shows almost constant curvature for the forward half, and a smooth gradual increase of curvature toward the stern (Fig. 8). Notice that the changes required were very slight 1/4 to 3/8 inch. (This demonstration of fairing the sheerline by refining the plan and profile views independently overlooks the fact that the sheerline is a 3D curve and will be viewed from other directions than horizontally and vertically. It has long been known to naval architects that achieving a pleasant 3D appearance of the sheerline (and other visually prominent lines) requires coordination between many views. It is much less widely known that the proper coordination can be stated as a simple geometric rule: besides being fair, the curve should lie in a single plane. We demonstrate making a planar sheerline in Tutorial 5 A Planar Sheerline. ) 4-22 Fairness of Curves and Surfaces December 1998

23 Fairness of Curves and Surfaces Fairness of Surfaces Fairing the profile line Similarly, let s check the profile line select profile ; Tools, Curvature Profile. 'P74' 'P64' 'P54' 'P44' 'P34' 'P24' 'P14' Fig. 9a Fig. 9b. Lat 0. This curvature profile reveals a bit of a hard upward turn at P44 and a nearly flat spot around P54. We can relieve the situation by raising P44 by.01 and lowering P54 by.031. Select P44 ; <Ctrl>+up arrow once (Z = -1.19). Select P54 ; <Ctrl>+down arrow 3 times; and if you re really fussy, <Shift>+<Ctrl>+down arrow once more, <Enter> (Z = -1.01; Fig. 10). [This should match our distribution file CLFT7X4A.MS2 (in the MSURFWIN\SAMPLES folder).] Fig. 10. Lat 0. Again, we made considerable improvement with only slight changes. Smoothing of curvature profiles can be practiced on many other design curves. Master curves should be checked and refined, especially when they have a large number of control points. In the past we have often checked the curvature profiles of Vertex Lines CCurves passing through the same vertex on each master curve, e.g. the second vertex or third vertex in this case. Fairness of Surfaces Simplicity is a major factor in constructing fair surfaces. The fewer control points you use for a spline curve, and the fewer master curves you use for a lofted surface, the more automatically the surface fairs. The other side of the coin is, of course, the fewer control points and master curves you use, the less control you have over what the curve or surface does in between its supports. A great deal can be done toward fairing hull surfaces by fairing the curves they are built from. Techniques for fairing curves using curvature profile displays were covered in the previous section. There are analogous tools for displaying the distributions of curvature measures over surfaces. These have considerable potential for revealing unfairness in surfaces. Surface curvatures Surfaces, having more complexity than curves, have more ways to measure their curvature. Definition 1: Normal Vector At a given point on a sufficiently smooth surface, there is a unique tangent plane. The normal vector is attached to the given point and is MultiSurf 4.0 Fairness of Curves and Surfaces 4-23

24 Fairness of Curves and Surfaces Fairness of Surfaces perpendicular to the tangent plane at that point (Fig. 11, left). See also Reference topic Defining Objects - orientation (normal orientation switch) on page 10. Definition 2: Normal Plane At a given point on a surface, any plane that passes through that point and includes the normal vector is called a normal plane (Fig. 11, right). There is an infinite family of normal planes at any given point. The members of this family can be distinguished by the angle A that they make with the v = constant snake through that point. Any of the normal planes intersects the surface. The intersection is a curve in the surface. A tangent plane normal planes v=constant Fig. 11. Normal vector (left); normal planes and normal curvature (right). Definition 3: Normal Curvature Normal curvature is the curvature of the curve created by intersecting the surface with a normal plane (Fig. 11, right). Normal curvature is a signed quantity. If the curve of intersection turns toward the positive normal direction, normal curvature is positive; if it turns away, normal curvature is negative. The units of normal curvature are 1/length, e.g. 1/ft if your model is in feet. The radius of curvature is the reciprocal of normal curvature. For instance, normal curvature of.03 is equivalent to that of a circle with a radius of units ( = 1/.03). Normal curvature depends on the orientation of the normal plane. As angle A varies, in fact, the variation of normal curvature is sinusoidal (Fig. 12), with (in general) a maximum and a minimum at angles A that are 90 degrees apart. (The exception is where normal curvature is constant with respect to A, such as any point on a sphere.) K 2 normal curvature 0 angle A K deg. 360 deg. Fig. 12. As angle A is changed, the normal curvature varies between the limits of the two principal curvatures. Definition 4: Principal Curvatures The maximum and minimum values of the normal curvature at a point are called the principal curvatures K 1, K 2 of the surface at that point. The units of principal curvature are 1/length, e.g. 1/ft if your model is in feet Fairness of Curves and Surfaces December 1998

25 Fairness of Curves and Surfaces Fairness of Surfaces Definition 5: Mean Curvature The average of the two principal curvature, (K 1 +K 2 )/2, is called the mean curvature of the surface at that point. The units of mean curvature are also 1/length. Definition 6: Gaussian Curvature The product of the two principal curvatures, K 1 x K 2, is called Gaussian curvature. Because K 1 and K 2 are signed quantities, so is Gaussian curvature. At a point on the surface where the principal curvatures have the same sign, Gaussian curvature is positive (Fig. 13, top). This type of surface is locally shaped like an ellipsoid or a sphere. At a point on the surface where the principal curvatures have opposite signs, the Gaussian curvature is negative (Fig. 13, bottom). This type of surface is locally shaped like a saddle or a potato chip. If one principal curvature is zero, the Gaussian curvature is also zero (Fig. 13, middle). If the Gaussian curvature is zero everywhere, the surface is developable, i.e. it can be rolled out flat onto a plane without stretching. Nonzero Gaussian curvature is often loosely referred to as compound curvature. K > 0 K = 0 K < 0 Fig. 13 Positive (sphere-like), zero (curved sheet), and negative (saddle-like) Gaussian curvature. The units of Gaussian curvature are 1/(length squared), e.g. 1/ft 2 if the model is in feet. Note: The principal, mean, and Gaussian curvatures are properties of the surface at a point. They are independent of any coordinates chosen to describe the surface and also of any snakes in the surface that happen to pass through the point. You can remove the coordinates, and remove any snakes in the surface, and these curvature properties remain unchanged. (The color displayed and the contours are independent of your choice of angle.) By contrast, the normal curvature at a point depends on the direction (angle) of the normal plane passing through the point. Some properties and applications for surface curvatures Gaussian curvature of zero is required for a ruled surface to be developable. The amount of Gaussian curvature quantifies how far a compound-curved surface is from developable, i.e. difficulty of forming from flat material. Negative Gaussian curvature indicates that there are hollows in some directions at that place on the surface. Smooth, fair surfaces generally show gentle color gradients for all these curvature measures. Local small bumps or hollows show up as abrupt color changes. Normal curvature is something like rocking a straightedge on the surface, along the direction of the selected line snakes. A zero-curvature place along such a line MultiSurf 4.0 Fairness of Curves and Surfaces 4-25

26 Fairness of Curves and Surfaces Fairness of Surfaces would be a place where the straightedge would bump or click because of a local flat spot or inflection. Examples MultiSurf offers normal, Gaussian, and mean surface curvature displays. However, the concept of normal curvature underlies all measures of surface curvature, and we feel normal curvature itself is the best tool for revealing flat spots, bumps, ridges, and other common surface irregularities that you would like to remove during design. A normal curvature display is a lot like rocking a straightedge on the surface in various orientations. We ll use two models to illustrate our discussion. We suggest you open them in MultiSurf and use Tools/ Surface Curvatures to look at the surface curvatures displays (they are rendered images that don t reproduce well in black-and-white and don t make nice pictures in the help file). Normal curvature First we ll look at CLFT7X4B (this is the CLFT7X4 model following fairing and planarizing of the sheerline; the.ms2 file is in the \MSURFWIN\SAMPLES directory). The u-direction for the hull runs in the direction of the master curves, from the sheerline to the profile; the tip of the bow is the 0,0 corner of the surface. Let s start by looking at normal curvature calculated at 0 degrees from the u-direction. (To do this: select the hull surface, then Tools/ Surface Curvatures/ Normal at 0 degrees.) This shows a shaded, color-coded map of normal curvatures of the surface, calculated along a family of parallel LineSnakes. For this selected angle (of zero from the u-direction), the snakes all run in the u-direction, i.e. along v = constant parametric lines. At any point on the surface, you cut the surface with a plane oriented normal (perpendicular) to the surface and tangent to the snake. The normal curvature is the curvature of this plane intersection. The normal curvature is positive if the snake is curving toward the positive normal direction. The numbers printed in color down the left-hand edge of the screen are the color code or key for the display. (See also: Program Operation - Menu Options - Tools/ Surface Curvatures.) This particular curvature display does not reveal any definite problems. Naturally the curvature is highest around the turn of the bilge, and near zero on the forward topsides. The display does reveal a small amount of concavity in the upper corner of the bow; the white line is the zero contour. The concavity is very small. It may be acceptable or desired; or it can be removed by moving the nearest interior control point, P22, outboard to Y = (To do this: switch to the wireframe view, select the point, then Edit/ Move Point.) Now let s look at normal curvature at 90 degrees from the u-direction (Tools/ Change Surface Curvatures/ Normal at 90 degrees). Here the linesnakes run along the v-direction, so the normal curvature is being measured in the direction of the lofting splines or longitudinals. In the topsides, this display shows a smooth progression from relatively low curvature at the bow, to higher curvature amidships, to lower curvature again at the stern, imitating the curvature distribution of the sheer. In the bottom, there are two irregularities: a flat spot (low curvature light cyan) near P53, and a slight incursion of higher curvature (dip of green) near P43. These can be smoothed by the following small changes: 4-26 Fairness of Curves and Surfaces December 1998