434 lines
13 KiB
Plaintext
434 lines
13 KiB
Plaintext
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INTRODUCTION
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============
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A sketch in SolveSpace consists of three basic elements: parameters,
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entities, and constraints.
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A parameter (Slvs_Param) is a single real number, represented internally
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by a double-precision floating point variable. The parameters are unknown
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variables that the solver modifies in order to satisfy the constraints.
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An entity (Slvs_Entity) is a geometric thing, like a point or a line
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segment or a circle. Entities are defined in terms of parameters,
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and in terms of other entities. For example, a point in three-space
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is represented by three parameters, corresponding to its x, y, and z
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coordinates in our base coordinate frame. A line segment is represented
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by two point entities, corresponding to its endpoints.
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A constraint (Slvs_Constraint) is a geometric property of an entity,
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or a relationship among multiple entities. For example, a point-point
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distance constraint will set the distance between two point entities.
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Paramters, entities, and constraints are typically referenced by their
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handles (Slvs_hParam, Slvs_hEntity, Slvs_hConstraint). These handles are
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32-bit integer values starting from 1. The zero handle is reserved. Each
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object has a unique handle within its type (but it's acceptable, for
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example to have a constraint with an Slvs_hConstraint of 7, and also to
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have an entity with an Slvs_hEntity of 7). The use of handles instead
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of pointers helps to avoid memory corruption.
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Entities and constraints are assigned into groups. A group is a set of
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entities and constraints that is solved simultaneously. In a parametric
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CAD system, a single group would typically correspond to a single sketch.
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Constraints within a group may refer to entities outside that group,
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but only the entities within that group will be modified by the solver.
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Consider point A in group 1, and point B in group 2. We have a constraint
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in group 2 that makes the points coincident. When we solve group 2, the
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solver is allowed to move point B to place it on top of point A. It is
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not allowed to move point A to put it on top of point B, because point
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A is outside the group being solved.
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This corresponds to the typical structure of a parametric CAD system. In a
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later sketch, we may constrain our entities against existing geometry from
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earlier sketches. The constraints will move the entities in our current
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sketch, but will not change the geometry from the earlier sketches.
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To use the solver, we first define a set of parameters, entities, and
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constraints. We provide an initial guess for each parameter; this is
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necessary to achieve convergence, and also determines which solution
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gets chosen when (finitely many) multiple solutions exist. Typically,
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these initial guesses are provided by the initial configuration in which
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the user drew the entities before constraining them.
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We then run the solver for a given group. The entities within that group
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are modified in an attempt to satisfy the constraints.
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After running the solver, there are three possible outcomes:
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* All constraints were satisfied to within our numerical
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tolerance (i.e., success). The result is equal to SLVS_RESULT_OKAY,
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and the parameters in param[] have been updated.
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* The solver can prove that two constraints are inconsistent (for
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example, if a line with nonzero length is constrained both
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horizontal and vertical). In that case, a list of inconsistent
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constraints is generated in failed[].
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* The solver cannot prove that two constraints are inconsistent, but
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it cannot find a solution. In that case, the list of unsatisfied
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constraints is generated in failed[].
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TYPES OF ENTITIES
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=================
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SLVS_E_POINT_IN_3D
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A point in 3d. Defined by three parameters:
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param[0] the point's x coordinate
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param[1] y
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param[1] z
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SLVS_E_POINT_IN_2D
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A point within a workplane. Defined by the workplane
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wrkpl
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and by two parameters
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param[0] the point's u coordinate
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param[1] v
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within the coordinate system of the workplane. For example, if the
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workplane is the zx plane, then u = z and v = x. If the workplane is
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parallel to the zx plane, but translated so that the workplane's
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origin is (3, 4, 5), then u = z - 5 and v = x - 3.
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SLVS_E_NORMAL_IN_3D
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A normal. In SolveSpace, "normals" represent a 3x3 rotation matrix
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from our base coordinate system to a new frame. Defined by the
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unit quaternion
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param[0] w
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param[1] x
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param[2] y
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param[3] z
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where the quaternion is given by w + x*i + y*j + z*k.
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It is useful to think of this quaternion as representing a plane
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through the origin. This plane has three associated vectors: basis
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vectors U, V that lie within the plane, and normal N that is
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perpendicular to it. This means that
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[ U V N ]'
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defines a 3x3 rotation matrix. So U, V, and N all have unit length,
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and are orthogonal so that
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U cross V = N
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V cross N = U
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N cross U = V
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Convenience functions (Slvs_Quaternion*) are provided to convert
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between this representation as vectors U, V, N and the unit
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quaternion.
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A unit quaternion has only 3 degrees of freedom, but is specified in
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terms of 4 parameters. An extra constraint is therefore generated
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implicitly, that
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w^2 + x^2 + y^2 + z^2 = 1
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SLVS_E_NORMAL_IN_2D
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A normal within a workplane. This is identical to the workplane's
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normal, so it is simply defined by
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wrkpl
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This entity type is used, for example, to define a circle that lies
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within a workplane. The circle's normal is the same as the workplane's
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normal, so we can use an SLVS_E_NORMAL_IN_2D to copy the workplane's
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normal.
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SLVS_E_DISTANCE
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A distance. This entity is used to define the radius of a circle, by
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a single parameter
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param[0] r
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SLVS_E_WORKPLANE
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An oriented plane, somewhere in 3d. This entity therefore has 6
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degrees of freedom: three translational, and three rotational. It is
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specified in terms of its origin
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point[0] origin
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and a normal
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normal
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The normal describes three vectors U, V, N, as discussed in the
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documentation for SLVS_E_NORMAL_IN_3D. The plane is therefore given
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by the equation
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p = origin + s*U + t*V
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for any scalar s and t.
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SLVS_E_LINE_SEGMENT
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A line segment between two endpoints
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point[0]
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point[1]
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SLVS_E_CUBIC
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A nonrational cubic Bezier segment
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point[0] starting point P0
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point[1] control point P1
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point[2] control point P2
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point[3] ending point P3
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The curve then has equation
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p(t) = P0*(1 - t)^3 + 3*P1*(1 - t)^2*t + 3*P2*(1 - t)*t^2 + P3*t^3
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as t goes from 0 to 1.
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SLVS_E_CIRCLE
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A complete circle. The circle lies within a plane with normal
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normal
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The circle is centered at
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point[0]
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The circle's radius is
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distance
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SLVS_E_ARC_OF_CIRCLE
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An arc of a circle. An arc must always lie within a workplane; it
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cannot be free in 3d. So it is specified with a workplane
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wrkpl
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It is then defined by three points
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point[0] center of the circle
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point[1] beginning of the arc
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point[2] end of the arc
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and its normal
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normal identical to the normal of the workplane
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The arc runs counter-clockwise from its beginning to its end (with
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the workplane's normal pointing towards the viewer). If the beginning
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and end of the arc are coincident, then the arc is considered to
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represent a full circle.
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This representation has an extra degree of freedom. An extra
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constraint is therefore generated implicitly, so that
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distance(center, beginning) = distance(center, end)
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TYPES OF CONSTRAINTS
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====================
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Many constraints can apply either in 3d, or in a workplane. This is
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determined by the wrkpl member of the constraint. If that member is set
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to SLVS_FREE_IN_3D, then the constraint applies in 3d. If that member
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is set equal to a workplane, the the constraint applies projected into
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that workplane. (For example, a constraint on the distance between two
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points actually applies to the projected distance).
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Constraints that may be used in 3d or projected into a workplane are
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marked with a single star (*). Constraints that must always be used with
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a workplane are marked with a double star (**). Constraints that ignore
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the wrkpl member are marked with no star.
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SLVS_C_PT_PT_DISTANCE*
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The distance between points ptA and ptB is equal to valA. This is an
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unsigned distance, so valA must always be positive.
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SLVS_C_PROJ_PT_DISTANCE
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The distance between points ptA and ptB, as projected along the line
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or normal entityA, is equal to valA. This is a signed distance.
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SLVS_C_POINTS_COINCIDENT*
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Points ptA and ptB are coincident (i.e., exactly on top of each
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other).
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SLVS_C_PT_PLANE_DISTANCE
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The distance from point ptA to workplane entityA is equal to
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valA. This is a signed distance; positive versus negative valA
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correspond to a point that is above vs. below the plane.
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SLVS_C_PT_LINE_DISTANCE*
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The distance from point ptA to line segment entityA is equal to valA.
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If the constraint is projected, then valA is a signed distance;
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positive versus negative valA correspond to a point that is above
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vs. below the line.
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If the constraint applies in 3d, then valA must always be positive.
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SLVS_C_PT_IN_PLANE
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The point ptA lies in plane entityA.
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SLVS_C_PT_ON_LINE*
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The point ptA lies on the line entityA.
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Note that this constraint removes one degree of freedom when projected
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in to the plane, but two degrees of freedom in 3d.
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SLVS_C_EQUAL_LENGTH_LINES*
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The lines entityA and entityB have equal length.
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SLVS_C_LENGTH_RATIO*
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The length of line entityA divided by the length of line entityB is
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equal to valA.
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SLVS_C_EQ_LEN_PT_LINE_D*
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The length of the line entityA is equal to the distance from point
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ptA to line entityB.
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SLVS_C_EQ_PT_LN_DISTANCES*
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The distance from the line entityA to the point ptA is equal to the
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distance from the line entityB to the point ptB.
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SLVS_C_EQUAL_ANGLE*
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The angle between lines entityA and entityB is equal to the angle
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between lines entityC and entityD.
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If other is true, then the angles are supplementary (i.e., theta1 =
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180 - theta2) instead of equal.
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SLVS_C_EQUAL_LINE_ARC_LEN*
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The length of the line entityA is equal to the length of the circular
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arc entityB.
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SLVS_C_SYMMETRIC*
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The points ptA and ptB are symmetric about the plane entityA. This
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means that they are on opposite sides of the plane and at equal
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distances from the plane, and that the line connecting ptA and ptB
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is normal to the plane.
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SLVS_C_SYMMETRIC_HORIZ
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SLVS_C_SYMMETRIC_VERT**
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The points ptA and ptB are symmetric about the horizontal or vertical
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axis of the specified workplane.
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SLVS_C_SYMMETRIC_LINE**
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The points ptA and ptB are symmetric about the line entityA.
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SLVS_C_AT_MIDPOINT*
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The point ptA lies at the midpoint of the line entityA.
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SLVS_C_HORIZONTAL
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SLVS_C_VERTICAL**
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The line connecting points ptA and ptB is horizontal or vertical. Or,
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the line segment entityA is horizontal or vertical. If points are
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specified then the line segment should be left zero, and if a line
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is specified then the points should be left zero.
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SLVS_C_DIAMETER
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The diameter of circle or arc entityA is equal to valA.
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SLVS_C_PT_ON_CIRCLE
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The point ptA lies on the right cylinder obtained by extruding circle
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entityA normal to its plane.
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SLVS_C_SAME_ORIENTATION
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The normals entityA and entityB describe identical rotations. This
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constraint therefore restricts three degrees of freedom.
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SLVS_C_ANGLE*
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The angle between lines entityA and entityB is equal to valA, where
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valA is specified in degrees. This constraint equation is written
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in the form
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(A dot B)/(|A||B|) = cos(valA)
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where A and B are vectors in the directions of lines A and B. This
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equation does not specify the angle unambiguously; for example,
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note that val A = +/- 90 degrees will produce the same equation.
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If other is true, then the constraint is instead that
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(A dot B)/(|A||B|) = -cos(valA)
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SLVS_C_PERPENDICULAR*
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Identical to SLVS_C_ANGLE with valA = 90 degrees.
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SLVS_C_PARALLEL*
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Lines entityA and entityB are parallel.
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Note that this constraint removes one degree of freedom when projected
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in to the plane, but two degrees of freedom in 3d.
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SLVS_C_ARC_LINE_TANGENT**
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The arc entityA is tangent to the line entityB. If other is false,
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then the arc is tangent at its beginning (point[1]). If other is true,
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then the arc is tangent at its end (point[2]).
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SLVS_C_CUBIC_LINE_TANGENT*
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The cubic entityA is tangent to the line entityB. If other is false,
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then the cubic is tangent at its beginning (point[0]). If other is
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true, then the arc is tangent at its end (point[3]).
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SLVS_C_EQUAL_RADIUS
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The circles or arcs entityA and entityB have equal radius.
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USING THE SOLVER
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================
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See the enclosed sample code, example.c.
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Copyright 2009 Useful Subset, LLC
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