Command Browser

This chapter is an alphabetical listing of all the commands provided by Descarta2D.  The syntax and usage of each command is described, as well as notes outlining special options and defaults.  Additionally, a cross-reference pointing to related commands is provided in each section.  Commands described as being low-level are used in the internal implementation of Descarta2D.  Low-level commands may be used freely, but they are not generally mentioned in the subject matter chapters of the book.

Angle2D [Top]

• Angle2D[line] computes the angle a line makes with the +x-axis, when measured counter-clockwise from the +x-axis to the line.

• Angles in Descarta2D are always specified and returned in radians.

• Angle2D[arc] computes the angular span of an arc.

• Angle2D[arc,t] computes the angle a radial diameter passing through the point at parameter t on the arc makes with the +x-axis.

• Angle2D[conic] returns the rotation angle of a conic curve.  The conic may be an ellipse, hyperbola or parabola.

• Angle2D[line,line] computes the angle measured counter-clockwise from the first line to the second line.    

• Angle2D[triangle,n] computes the vertex angle at vertex n of a triangle.

• See also: PrimaryAngle2D, PrimaryAngleRange2D, ReflectAngle2D.

Apex2D [Top]

• Apex2D is a keyword indicating the construction of the apex control point of a conic arc.

• See also: Point2D.

Arc2D [Top]

• Arc2D[{,},{,},B] is the standard representation of an arc.  The coordinates of the start point are and the coordinates of the end point are .  The bulge factor is the positive number B.  The arc is traversed counter-clockwise from the start point to the end point.

• The bulge factor, B, is the ratio of the arc's height, h, to half the chord length, d/2; so B=2h/d.

• Arc2D[{,},{,},B][t] and arc[t] return the {x,y} coordinates of a point at parameter t on an arc.  Parameter values in the range 0≤t≤1 produce coordinates covering the complete span of the arc.

• Arc2D[{,},{,},B][{,}] produces graphics primitives for a portion of an arc between parameters and when plotted.

• Arc2D[arc,Complement2D] constructs the complement of an arc.

• Arc2D[point,r,{,}] constructs an arc from the center point, radius and span.  The angles, and , are measured counter-clockwise from the +x-axis.

• Arc2D[point,r,{,}] constructs an arc from the center point, radius and the start and end points of the span.  The start/end points do not need to lie on the arc, although they cannot be coincident with the center.

• Arc2D[point,point,point] constructs an arc passing through three points.  The first and third points define the span of the arc.

• Arc2D[{point,θ},point] constructs an arc from a start point with entry angle and an end point.

• See also: Bulge2D, Complement2D.

ArcLength2D [Top]

• ArcLength2D[curve,{,}] computes the arc length of a curve between two parameter values.

• The curve may be an arc, circle, ellipse, hyperbola, line, line segment or parabola.

• N[ArcLength2D[cnarc,{,}]] numerically computes the arc length of a conic arc between two parameter values.

• See also: Circumference2D, Length2D, Perimeter2D, Span2D.

Area2D [Top]

• Area2D[curve] computes the area associated with a curve.

• The curve may be an arc, circle, conic arc, ellipse or triangle.

• Area2D[arc] computes the area between an arc and its chord.

• Area2D[circle] computes the area of a circle.

• Area2D[cnarc] computes the area between a conic arc and its chord.

• Area2D[ellipse] computes the area of an ellipse.

• Area2D[triangle] computes the area of a triangle.

• See also: SectorArea2D, SegmentArea2D.

AskCurveLength2D [Top]

• AskCurveLength2D[] is a low-level function that returns the value of the CurveLength2D option of the Sketch2D command.

• See also: CurveLength2D, Sketch2D.

Asymptotes2D [Top]

• Asymptotes2D[hyperbola] constructs a list containing the two asymptote lines of a hyperbola.

Bulge2D [Top]

• Bulge2D[arc] returns the bulge factor of an arc.

• See also: Arc2D.

Centroid2D [Top]

• Centroid2D is a keyword indicating the construction of a triangle's centroid point.

• See also: Point2D.

ChopImaginary2D [Top]

• ChopImaginary2D[expr,tol] is a low-level function that removes insignificant imaginary parts of complex numbers in an expression.  The imaginary part is considered insignificant if its absolute value is less than the tolerance.

• The tolerance, if omitted, defaults to .

Circle2D [Top]

• Circle2D[{h,k},r] is the standard representation of a circle.  The coordinates of the center point of the circle are {h,k} and the radius is r.

• Circle2D[{h,k},r}][θ] and circle[θ] return the {x,y} coordinates of a point at parameter θ on a circle.  Parameter values in the range 0≤θ<2π produce coordinates covering the complete circumference of the circle.

• Circle2D[{h,k},r}][{,}] produces graphics primitives for the arc of the circle between parameters and when plotted.

• Circle2D[arc] constructs the circle underlying an arc.

• Circle2D[circle,circle,k,Pencil2D] constructs a circle, parameterized by the variable k, that represents the family (pencil) of circles passing through the intersection points of the two given circles.  The family of circles is valid even if the two circles do not intersect as they will share a common radical axis.

• Circle2D[lnseg] constructs the circle whose diameter chord is a given line segment.

• Circle2D[point,r] constructs the circle centered at a point with a given radius.

• Circle2D[point,point] constructs the circle given a center point and a point on the circle.

• Circle2D[point,point,point] constructs a circle through three points.

• Circle2D[point,line] constructs a circle with a given center point and tangent to a line.

• Circle2D[quad] constructs the circle associated with a quadratic.

• Circle2D[triangle,Circumscribed2D] constructs a circle circumscribed about a triangle.

• Circle2D[triangle,Inscribed2D] constructs a circle inscribed inside a triangle.

• See also: Inscribed2D, Circumscribed2D, Pencil2D, TangentCircles2D.

Circumference2D [Top]

• Circumference2D[circle] computes the circumference of a circle.

• Circumference2D[ellipse] computes the circumference of an ellipse.

• See also: ArcLength2D.

Circumscribed2D [Top]

• Circumscribed2D is a keyword indicating a construction involving a triangle's circumscribed circle.

• See also: Circle2D, Point2D.

Complement2D [Top]

• Complement2D is a keyword indicating the construction of an arc's complement.

• See also: Arc2D.

ConicArc2D [Top]

• ConicArc2D[{,},{,},{,},ρ] is the standard representation of a conic arc.  The coordinates of the start point are {,}, the coordinates of the apex point are {,} and the coordinates of the end point are {,}.  The projective discriminant is ρ.

• ConicArc2D[{,},{,},{,},ρ][t] and cnarc[t] return the {x,y} coordinates of a point at parameter t on a conic arc.  Parameter values in the range 0≤t≤1 produce coordinates covering the entire length of the conic arc.

• ConicArc2D[{,},{,},{,},ρ][{,}] produces graphics primitives representing the portion of the conic arc between parameters and when plotted.

• ConicArc2D[line,conic] constructs a conic arc defined by a conic (circle, ellipse, hyperbola or parabola) and a line containing the conic arc's chord.

Conjugate2D [Top]

• Conjugate2D is a keyword indicating the construction of a conjugate hyperbola.

• See also: Hyperbola2D.

Coordinates2D [Top]

• Coordinates2D[args..] returns the {x,y} coordinates of the point that would be returned by the function Point2D[args..].

• See also: Point2D, XCoordinate2D, YCoordinate2D.

CurveLength2D [Top]

• CurveLength2D is an option for the Sketch2D command specifying the approximate length that an unbounded curve should be rendered when plotted.

• The initial default, if not specified, is 10.  The default can by changed using the Mathematica SetOptions command.

• See also: AskCurveLength2D, Sketch2D.

CurveLimits2D [Top]

• CurveLimits2D[coords,curve] is a low-level function that computes a list of two parameter values on a curve such that the point whose coordinates are given is a distance CurveLength2D/2 from the points on the curve at the parameter values.

• See also: AskCurveLength2D, CurveLength2D, Sketch2D.

Directrices2D [Top]

• Directrices2D[conic] returns a list of the directrix line(s) of a conic curve.

• The conic may be an ellipse, hyperbola or parabola.  If the conic is an ellipse or hyperbola there are two directrix lines in the list; if the conic is a parabola there is one directrix line in the list.

Distance2D [Top]

• Distance2D[coords,coords] computes the distance between two positions given by coordinates.

• Distance2D[point,point] computes the distance between two points.

• Distance2D[point,line] computes the distance between a point and a line.

• Distance2D[point,circle] computes the distance between a point and a circle.

Eccentricity2D [Top]

• Eccentricity2D[conic] computes the eccentricity of a conic.

• The conic may be a ellipse, hyperbola or parabola.

Ellipse2D [Top]

• Ellipse2D[{h,k},a,b,θ] is the standard representation of an ellipse.  The coordinates of the center point are {h,k}, the length of the semi-major axis is a, the length of the semi-minor axis is b and the angle of rotation, counter-clockwise with respect to the +x-axis, is θ.

• ] and ellipse[] return the {x,y} coordinates of a point at parameter on an ellipse.  Parameter values in the range produce coordinates covering the complete circumference of the ellipse.

• Ellipse2D[{h,k},a,b,θ][{,}] produces graphics primitives on the portion of the ellipse between parameter values and when plotted.

• Ellipse2D[point,line,e] constructs an ellipse defined by a focus point, directrix line and eccentricity.

• Ellipse2D[point,point,e] constructs an ellipse from two focus points and the eccentricity.

• Ellipse2D[{point,point},e] constructs an ellipse from two vertex points and the eccentricity.

Equation2D [Top]

• Equation2D[line,{x,y}] returns the equation A x+B y+C==0, which is the equation of the line.

• Equation2D[quad,{x,y}] returns which is the equation of the quadratic.

• See also: Polynomial2D.

FocalChords2D [Top]

• FocalChords2D[conic] returns a list containing the focal chords of a conic curve (line segments).

• The conic may be an ellipse, hyperbola or parabola.  If the conic is an ellipse or hyperbola the list contains two focal chords (line segments); if the conic is a parabola the list contains a single focal chord (line segment).

FocalLength2D [Top]

• FocalLength2D[parabola] returns the focal length of a parabola.

• See also: Parabola2D.

Foci2D [Top]

• Foci2D[conic] returns a list containing the focus point(s) of a conic.

• The conic may be an ellipse, hyperbola or parabola.  If the conic is an ellipse or hyperbola the list contains two focus points; if the conic is a parabola the list contains a single focus point.

Hyperbola2D [Top]

• Hyperbola2D[{h,k},a,b,θ] is the standard representation of a hyperbola.  The coordinates of the center point are {h,k}, the length of the semi-transverse axis is a, the length of the semi-conjugate axis is b and the angle of rotation, counter-clockwise with respect to the +x-axis, is θ.

• Hyperbola2D[{h,k},a,b,θ][t] and hyperbola[t] return the {x,y} coordinates of a point at parameter t on the primary branch of a hyperbola.  Parameter values in the range -∞<t<+∞ cover the complete hyperbola branch.  The primary branch opens about the +x-axis when the angle of rotation is zero.

• Hyperbola2D[{h,k},a,b,θ,True][t] returns the {x,y} coordinates of a point at parameter t on the non-primary (reflected) branch of a hyperbola (used only for graphics rendering).

• Hyperbola2D[{h,k},a,b,θ][{,}] produces graphics primitives for a portion of the hyperbola between parameters values and when plotting.  If the parameters represent a portion of the primary branch of the hyperbola; if the parameters represent a portion of the other branch.

• Hyperbola2D[hyperbola,Conjugate2D] constructs the conjugate of a hyperbola.

• Hyperbola2D[point,line,e] constructs a hyperbola defined by a focus point, directrix line and eccentricity.

• Hyperbola2D[point,point,e] constructs a hyperbola from two focus points and the eccentricity.

• Hyperbola2D[{point,point},e] constructs a hyperbola from two vertex points and the eccentricity.

• See also: Conjugate2D.

Inscribed2D [Top]

• Inscribed2D is a keyword indicating a construction involving a triangle's inscribed circle.

• See also: Circle2D, Point2D.

Is2D [Top]

• Is2D[object,objHeadList] is a low-level function that returns True if the object is a valid Descarta2D object and its head is included in the head list; otherwise, returns False.

IsApproximate2D [Top]

• IsApproximate2D[expr] is a low-level function that returns True if the expression contains approximate real numbers; otherwise, returns False.

• The function will attempt to detect if the pending evaluation will eventually be approximated using the N[expr] function.  If this condition is detected the function will also return True.

IsCoincident2D [Top]

• IsCoincident2D[obj,obj] returns True if two objects are of the same type and are coincident; otherwise, returns False.  The objects may be circles, coordinates, lines, points or quadratics.

• The function returns unevaluated if the two objects are of a different type.

• IsCoincident2D[objList] returns True if any pair of objects in a list are of the same type and are coincident; otherwise, returns False.

IsCollinear2D [Top]

• IsCollinear2D[point,point,point] returns True if three points are collinear; otherwise, returns False.

• IsCollinear2D[ptList] returns True if any triple of points in a list is collinear; otherwise, returns False.

IsComplex2D [Top]

• IsComplex2D[expr,tol] is a low-level function that returns True if the expression, when evaluated, contains a complex number (a number is considered complex if the absolute value of its imaginary part is greater than the tolerance); otherwise, returns False.

• The tolerance, if omitted, defaults to .

• IsComplex2D[exprList,tol] returns True if any expression in a list, when evaluated, contains a complex number; otherwise, returns False.

• IsComplex2D[exprList,Or,tol] returns True if any expression in a list, when evaluated, contains a complex number; otherwise, returns False.

• IsComplex2D[exprList,And,tol] returns True if all the expressions in a list, when evaluated, contain complex numbers; otherwise, returns False.

IsConcentric2D [Top]

• IsConcentric2D[cirle,circle] returns True if two circles are concentric; otherwise, returns False.

• IsConcentric2D[cirList] returns True if any pair of circles in a list are concentric; otherwise, returns False.

IsConcurrent2D [Top]

• IsConcurrent2D[line,line,line] returns True if three lines are concurrent (intersect in a common point); otherwise, returns False.

• IsConcurrent2D[lnList] returns True if any triple of lines in a list is concurrent; otherwise, returns False.

IsDisplay2D [Top]

• IsDisplay2D[object] is a low-level function that returns True if the object is a displayable Descarta2D object; otherwise, returns False.

IsNegative2D [Top]

• IsNegative2D[expr,tol] is a low-level function that returns True if the expression, when evaluated, is negative (a number is considered negative if it is less than zero and its absolute value is greater than the tolerance); otherwise, returns False.

• The tolerance, if omitted, defaults to .

• IsNegative2D[exprList,tol] returns True if any expression in a list, when evaluated, is negative; otherwise, returns False.

• IsNegative2D[exprList,Or,tol] returns True if any expression in a list, when evaluated, is negative; otherwise, returns False.

• IsNegative2D[exprList,And,tol] returns True if all the expressions in a list, when evaluated, are negative; otherwise, returns False.

• See also: IsZero2D, IsZeroOrNegative2D.

IsNumeric2D [Top]

• IsNumeric2D[expr,tol] is a low-level function that returns True if all the atoms in an expression can be evaluated to real numbers (a complex number is considered real if the absolute value of its imaginary part is less than the tolerance); otherwise, returns False.

• The tolerance, if omitted, defaults to .

• IsNumeric2D[expr,funcName,tol] returns True if all the atoms in an expression can be evaluated to real numbers; otherwise, returns False and displays a message stating that the function, funcName, requires numerical arguments.  This form is a low-level function and is intended to be used for argument checking.

IsOn2D [Top]

• IsOn2D[point,curve] returns True if a point is on a curve; otherwise, returns False.

• The curve may be a line, circle or quadratic.

• IsOn2D[point,Quadratic2D[conic]] returns True if a point is on a conic; otherwise, returns False.  The conic may be a circle, ellipse, hyperbola or parabola.

IsParallel2D [Top]

• IsParallel2D[line,line] returns True if two lines are parallel; otherwise, returns False.

• IsParallel2D[lnList] returns True if any pair of lines in a list is parallel; otherwise, returns False.

• See also: IsTripleParallel2D.

IsPerpendicular2D [Top]

• IsPerpendicular2D[line,line] returns True if two lines are perpendicular; otherwise, returns False.

• IsPerpendicular2D[lnList] returns True if any pair of lines in a list is perpendicular; otherwise, returns False.

IsReal2D [Top]

• IsReal2D[expr,tol] is a low-level function that returns True if the expression, when evaluated, is a real number (a complex number is considered real if the absolute value of its imaginary part is less than the tolerance); otherwise, returns False.

• The tolerance, if omitted, defaults to .

IsScalar2D [Top]

• IsScalar2D[expr] is a low-level function that returns True if the expression appears to be a scalar quantity—that is, it cannot be recognized as a list, a complex number or a Descarta2D object; otherwise, returns False.

• This function is used by Descarta2D for argument checking.

• See also: IsScalarPair2D.

IsScalarPair2D [Top]

• IsScalarPair2D[{expr,expr}] is a low-level function that returns True if both expressions appear to be scalar quantities—that is, they cannot be recognized as lists, complex numbers or Descarta2D objects; otherwise, returns False.

• This function is used by Descarta2D for argument checking.

• See also: IsScalar2D.

IsTangent2D [Top]

• IsTangent2D[line,circle] returns True if a line is tangent to a circle; otherwise, returns False.

• IsTangent2D[line,quad] returns True if a line is tangent to a quadratic; otherwise, returns False.

• IsTangent2D[line,Quadratic2D[conic]] returns True if a line is tangent to a conic; otherwise, returns False.  The conic may be a circle, ellipse, hyperbola or parabola.

• IsTangent2D[circle,circle] returns True if two circles are tangent to each other; otherwise, returns False.

IsTinyImaginary2D [Top]

• IsTinyImaginary2D[expr,tol] is a low-level function that returns True if any complex number in an expression has a tiny imaginary part (the imaginary part is considered tiny if its absolute value is less than the tolerance); otherwise, returns False.

• The tolerance, if omitted, defaults to .

IsTripleParallel2D [Top]

• IsTripleParallel2D[line,line,line] returns True if three lines are mutually parallel; otherwise, returns False.

• IsTripleParallel2D[lnList] returns True if any triple of lines in a list is mutually parallel; otherwise, returns False.

• See also: IsParallel2D.

IsValid2D [Top]

• IsValid2D[object] is a low-level function that returns True if the object is syntactically valid; otherwise, returns False.

• The object may be an arc, circle, conic arc, ellipse, hyperbola, line, line segment, parabola, point, quadratic or triangle.

IsZero2D [Top]

• IsZero2D[expr,tol] is a low-level function that returns True if the expression, when evaluated, is zero (a number is considered zero if its absolute value is less than the tolerance); otherwise, returns False.

• The tolerance, if omitted, defaults to .

• IsZero2D[exprList,tol] returns True if any expression in a list, when evaluated, is zero; otherwise, returns False.

• IsZero2D[exprList,Or,tol] returns True if any expression in a list, when evaluated, is zero; otherwise, returns False.

• IsZero2D[exprList,And,tol] returns True if all the expressions in a list, when evaluated, are zero; otherwise, returns False.

• See also: IsNegative2D, IsZeroOrNegative2D.

IsZeroOrNegative2D [Top]

• IsZeroOrNegative2D[expr,tol] returns True if the expression, when evaluated, is zero or negative; otherwise, returns False.

• The tolerance, if omitted, defaults to .

• IsZeroOrNegative2D[exprList,tol] returns True if any expression in a list, when evaluated, is zero or negative; otherwise, returns False.

• IsZeroOrNegative2D[exprList,Or,tol] returns True if any expression in a list, when evaluated, is zero or negative; otherwise, returns False.

• IsZeroOrNegative2D[exprList,And,tol] returns True if all the expressions in a list, when evaluated, are zero or negative; otherwise, returns False.

• See also: IsNegative2D, IsZero2D.

Length2D [Top]

• Length2D[lnseg] computes the length of a line segment.

• See also: ArcLength2D.

Line2D [Top]

• Line2D[A,B,C] is the standard representation of the line A x+B y+C=0.

• Line2D[A,B,C][t] and line[t] return the {x,y} coordinates of a point at parameter t on a line.  Parameter values in the range -∞<t<+∞ produce coordinates covering the complete line.

• Line2D[A,B,C][{,}] produces graphics primitives for the line segment between parameters and when plotting.

• Line2D[circle,circle] constructs the radical axis line of two circles.

• Line2D[coords,coords] constructs a line through two positions specified by {x,y} coordinates.

• Line2D[ellipse] constructs a line which contains the major axis of an ellipse.

• Line2D[eqn,{x,y}] constructs a line from the equation A x+B y+C==0.

• Line2D[hyperbola] constructs a line which contains the transverse axis of a hyperbola.

• Line2D[line] constructs a line with normalized coefficients.

• Line2D[line,d] constructs a line offset a distance d from a given line.  The distance may be positive or negative producing one of two possible offsets.

• Line2D[line,line,k,Pencil2D] constructs a family of lines (pencil), parameterized by k, passing through the intersection point of two given lines.

• Line2D[lnseg] constructs a line containing a line segment.

• Line2D[lnseg,Perpendicular2D] constructs a line that is the perpendicular bisector of a line segment.

• Line2D[parabola] constructs a line which contains the axis of a parabola.

• Line2D[point,curve] constructs the polar (line) of a curve given the pole (point).  If the pole (point) is on the curve, then the polar (line) is the tangent to the curve at the pole (point).  The curve may be a circle, ellipse, hyperbola, parabola or quadratic.

• Line2D[point,k,Pencil2D] constructs a family of lines (pencil), parameterized by k, passing through a point.

• Line2D[point,line] constructs a line through a point perpendicular to a line.

• Line2D[point,line,Perpendicular2D] also constructs a line through a point perpendicular to a line.

• Line2D[point,line,Parallel2D] constructs a line through a point parallel to a line.

• Line2D[point,m] constructs a line with slope m passing through a point.

• Line2D[point,Infinity] constructs a vertical line through a point.

• Line2D[point,point] constructs a line through two points.

• Line2D[point,point,Perpendicular2D] constructs a line equidistant from two points.  This line is the perpendicular bisector of the line segment defined by the two points.

• Line2D[poly,{x,y}] constructs a line from the polynomial A x+B y+C.

• Line2D[triangle,,] constructs a line containing vertices and of a triangle.

• See also: Parallel2D, Pencil2D, Perpendicular2D.

Loci2D [Top]

• Loci2D[quad] returns a list of objects represented by a quadratic.  The list may contain a conic, one or two lines, a point or it may be empty.

• Loci2D[cnarc] returns a list containing the curve underlying a conic arc.

• Loci2D[point,length,e] returns a list containing the conic defined by the vertex equation parameters.  The point is the vertex point, the length is the focal length and the constant, e, is the eccentricity.  The conic is constructed in standard position.

• Loci2D[point,length,e,θ] returns a list containing the conic defined by the vertex equation parameters.  The point is the vertex point, the length is the focal length, the constant, e, is the eccentricity and θ is the angle of rotation.

• Loci2D[point,line,e] returns a list containing the conic defined by a focus point, directrix line and eccentricity.

MakePrimitives2D [Top]

• MakePrimitives2D[curve,{,}] is a low-level function that returns a list of Mathematica graphics primitives approximating a curve between two parameter values.

• The curve may be an arc, circle, conic arc, ellipse, hyperbola, line, line segment or parabola.

MaxSeconds2D [Top]

• MaxSeconds2D is a keyword indicating the maximum number of seconds allowed for solving equations.

• See also: Solve2D.

MedialEquations2D [Top]

• MedialEquations2D[{obj,obj}] returns a list of lines or quadratics equidistant from two given objects.  The given objects may be points, lines or circles.

• See also: MedialLoci2D.

MedialLoci2D [Top]

• MedialLoci2D[{obj,obj}] returns a list of objects equidistant from two given objects.  The given objects may be points, lines or circles.

• See also: MedialEquations2D.

ObjectNames2D [Top]

• ObjectNames2D[] returns a list of strings which are the names of all the Descarta2D objects.

Parabola2D [Top]

• Parabola2D[{h,k},f,θ] is the standard representation of a parabola.  The coordinates of the vertex point are {h,k}, the focal length is f and the angle of rotation, counter-clockwise with respect to the +x-axis, is θ.

• Parabola2D[{h,k},f,θ][t] and parabola[t] return the {x,y} coordinates of a point at parameter t on a parabola.  Parameter values in the range -∞<t<+∞ produce coordinates covering the complete parabola.

• Parabola2D[{h,k},f,θ][{,}] produces graphics primitives for the portion of the parabola between parameters and when plotting.

• Parabola2D[point,line] constructs a parabola defined by a focus point and a directrix line.

Parallel2D [Top]

• Parallel2D is a keyword indicating a parallel construction.

• See also: Line2D, TangentLines2D.

Parameters2D [Top]

• Parameters2D[line,curve] computes a list of the two parameters where a line intersects a curve.