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Arcs

Definitions

Bulge Factor Arc

Three-Point Arc

Parametric Equations

Points and Angles at Parameters

Arcs from Ray Points

Explorations

We continue our study of circles by focusing on bounded portions of a circle's circumference commonly called arcs. Many of the interesting properties of arcs arise when considering how their end points and slopes meet with other curves. For example, many mechanical artifacts use arcs to construct transitions between the primary faces of the object giving a smoother and more durable design.

In addition to the topics presented in this chapter, a subsequent chapter will discuss another interesting use of arcs, the so-called biarc configuration of two arcs used to blend curves together smoothly.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Definitions [Top]

Definition of an arc.

Consider the parametric equations of a circle

x=h+rcosθ and y=k+rsinθ

where the point C(h,k) is the center of the circle and r is the radius of the circle. A circle is defined to be the set of points P(x,y) for all values of θ such that 0≤θ<2π (radians). Using the same parametric equations, a circular arc may be defined to be the set of points P(x,y) for all values of θ such that , where and . The point where is called the start point of the arc, and the point where is called the end point of the arc. The angle the directed line makes with the +x-axis is called the start angle of the arc; the angle the directed line makes with the +x-axis is called the end angle of the arc. The center point C(h,k) of the circle is also the center point of the arc, and the radius, r, of the circle is the radius of the arc.

Let and be the lines determined by the center point C and the start and end point of the arc, respectively. The angle between lines and is called the angular span of the arc. An arc with an angular span of π radians () is called a semicircle. The area bounded by line segments and and the arc itself is called a sector. The area bounded by the arc and is called a segment and the line segment itself is called the chord of the arc.

Example: Plot the arc centered at the point (-2,1) with a radius 6 and start angle of π/6 radians and end angle of π/2 radians. Include the center point of the arc in the plot.

Solution: Circle2D[{h,k},r][{,}] represents an arc of a circle between parameters and when plotting.

p1=Point2D[{-2,1}];

a1=Circle2D[{-2,1},6][{Pi/6,Pi/2}];

Sketch2D[{p1,a1}]

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Bulge Factor Arc [Top]

We now consider an arc representation involving the arc's start and end points, the so-called bulge factor arc as illustrated in Figure [arc:fig09]. A bulge factor arc is specified by its start and end points plus an additional number specifying the "bulge" (or fatness) of the desired arc. More precisely, if and are the start and end points of the arc, and is the midpoint of the arc, then the bulge factor, B, is defined to be the (non-zero, positive) ratio

where D is the distance between and and H is the distance from to the chordal line defined by and . Thus, an arc with B=1 will be a semicircle. Closer examination of the definition of the bulge factor arc reveals that for a given value of B there are two arcs satisfying the definition. These arcs are mirror images of each other (the line passing through and is the reflection line). To distinguish between these two arcs we make the arbitrary definition that the arc will be traversed counter-clockwise from to . The mirrored arc is represented by interchanging the roles of and .

Bulge factor arc definition.

Radius and Center

In order to determine defining parameters of the circle underlying the bulge factor arc, we need to determine the radius, r, and the center point, C(h,k), in terms of the points, and , and the bulge factor, B. Consider the right triangle where is the midpoint of the chord . By the Pythagorean Theorem

or

Solving for r and substituting H=BD/2 yields

where r>0, since B>0 and D>0.

To find the coordinates C(h,k) of the center point of the arc we note that the center is offset from point a distance (r-H). The direction of the offset is rotated -90 degrees from the vector . Therefore, the equations for C(h,k) are

where

It is clear from the expressions for r and C that if we replace B with 1/B we get an arc with the same radius and center, whose locus is counter-clockwise from to . This arc is the complement of the original arc. The reflection of the original arc in the chord may be obtained by reversing the roles of and and using the same value, B, as the bulge factor.

Example: Plot the arc with end points (1,0) and (0,1) with a bulge factor of 1/2. Find the mirror image of the arc reflected in the chord.

Solution: The standard representation of an arc in Descarta2D is

Arc2D[coords,coords,B]

where the start and end point coordinates are given as the first two arguments, respectively, and the bulge factor is the third argument. The arc reflected in the chord is constructed by reversing the roles of the start and end points.

a1=Arc2D[{1,0},{0,1},1/2];

a2=Arc2D[{0,1},{1,0},1/2];

Sketch2D[{a1,a2,Point2D[{1,0}],Point2D[{0,1}]}]

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Descarta2D Hint: The Descarta2D function Arc2D[arc,Complement2D] constructs the complement of an arc.

Angles

Let θ be the angular span of a bulge factor arc defined by points and and bulge factor B. Once again examining the right triangle reveals that

= | ||

= |

Using the trigonometric identity

we find that

From this equation it is clear that if B<1, the arc is a minor arc whose angular span is in the range 0<θ<π; if B>1 the arc is a major arc with π<θ<2π.

Let α denote the angle between the initial tangent vector, , and the chord vector, , considered positive when is clockwise from the chord vector, and negative when is counter-clockwise from the chord. Note that -π<α<π and |α|=θ/2. Therefore, B=tan(α/2).

From Equation ([arc:eqn01]) we can derive an expression for B in terms of sinα and cosα as follows

Solving this quadratic for B in terms of s and c yields

B | = |

(The positive root of the quadratic is selected in order to insure that B has the same sign as s. If B turns out to be negative, then the arc's start and end points are interchanged and the absolute value of B is the positive bulge factor.) The constants s and c are some multiple of sinα and cosα and immediately provide several useful techniques for constructing arcs. These techniques are illustrated in the "Explorations" section at the end of this chapter.

Three-Point Arc [Top]

Let and be the start and end points of an arc and let point P be any other point on the arc. One method for constructing the arc through the three points is to first construct the underlying circle through the three points and then compute the limiting angles of the arc from the end points and the center. Alternately, the bulge factor arc form provides an appealing method for computing the arc. Note that and are the chord end points required in the bulge factor arc formulation and in order to fully define the arc, we need to determine its bulge factor, B. As the third point P traverses the arc the angle subtended at P by the chord remains constant at the value (π-α). Thus, using the simpler vector form,

s | = | |

c | = |

From s and c we compute B using Equation ([arc:eqn02]).

Example: Find and plot the arc passing through the points (4,2), (-2,4), and (0,-6).

Solution: The function Arc2D[point,point,point] returns an arc through three points. The first and third points are assumed to be the end points of the arc chord.

p1=Point2D[{4,2}];

p2=Point2D[{-2,4}];

p3=Point2D[{0,6}];

a1=Arc2D[p1,p2,p3] //N

Sketch2D[{p1,p2,p3,a1}]

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Parametric Equations [Top]

One possible set of parametric equations for an arc is very similar to those of a circle since they both have the same underlying curve. The parameter, t, can be scaled in a different manner so that parameter value t=0 produces the start point of the arc, and parameter value t=1 produces the end point. The resulting parametric equations are

x | = | |

y | = |

where (h,k) is the center of the arc, r is the radius, and and are the start and end angles, respectively, of the arc.

Alternatively, since the standard form of an arc used in Descarta2D is

Arc2D[{,},{,},B],

we seek parametric equations involving only the start and end point coordinates and the bulge factor. The equations are given by

x | = | |

y | = |

where is the span of the arc and C(h,k) is the center point of the arc. Parameter values in the range 0≤t≤1 generate coordinates covering the span of the arc.

Example: Plot eight equally spaced points on the arc between the points (-3,2) and (2,1) with a bulge factor of 3/2

Solution: The Descarta2D function Arc2D[{,},{,},B][t] returns the coordinates at parameter value t on the arc.

a1=Arc2D[{-3,2},{2,1},3/2];

pts=Map[Point2D[a1[#]]&,Range[0,7]/7];

Sketch2D[{a1,pts}]

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Points and Angles at Parameters [Top]

Using the parametric equations of an arc defined in the previous section we can find the coordinates of any point on the arc corresponding to an angle θ in the range . The parametric equations for an arc as defined in Descarta2D are normalized so that the parameter value 0 generates the start point of the arc and the parameter value 1 generates the end point of the arc. Parametric values t, 0<t<1, will generate points on the arc between the start and end points.

Example: For the arc between the points and with a bulge factor of 3/2, use the parametric definition of an arc to find and plot the start point, end point and midpoint of the arc.

Solution: The function Arc2D[{,},{,},B][t] returns a list of coordinates representing the point on the arc at a given parameter value.

a1=Arc2D[{-2,1},{2,2},3/2];

coords={a1[0],a1[1/2],a1[1]} //N

Sketch2D[{a1, Map[Point2D,coords]}]

Whereas the Descarta2D function arc[t] generates the coordinates of the point on the arc at parameter t, the function Angle2D[arc,t] returns the angle (in radians) on the arc at parameter t with respect to a horizontal line such as the x-axis.

Angle2D[a1,1/2] //N

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Arcs from Ray Points [Top]

Sometimes it is more convenient to specify the start and end points of an arc, rather than the start and end angles. One obvious construction method is to specify the center and radius along with the start and end points. Let and be the desired start and end points of an arc centered at C(h,k) with radius r. Using simple trigonometry, the start and end angles of the arc are given by

The arc tangent function used to implement these equations must be sophisticated enough to assign the proper angle based on which quadrant the points are located in. Mathematica provides such an arc tangent function that takes the numerator and denominator as separate arguments and computes the angle in the proper quadrant.

Example: Plot the arc centered at the point (2,1) with radius 1, and with start and end points of (7,1) and (2,6).

Solution: The function Arc2D[point,r,{point,point}] returns a bulge factor arc given the center point, radius, and start and end points.

a1=Arc2D[p0=Point2D[{2,1}],5,

{p1=Point2D[{7,1}],p2=Point2D[{2,6}]}]

Sketch2D[{a1,p0,p1,p2},

PlotRange->{{0,8},{0,8}}]

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Descarta2D Hint: The Arc2D function introduced in the previous example allows more flexible input of the arc start and end points than is obvious from the example. These points may be located at any position on the ray extending from the center point of the arc to the desired arc start and end points. The arc will be bounded by the points of intersection between the circle underlying the arc and the rays defined from the center point to the specified start and end points.

Explorations [Top]

Arc from Bounding Points and Entry Direction. [arcentry.html]

Let and be the start and end points, respectively, of an arc and P be a third point on the vector tangent to the arc at . Show that

s | = | |

c | = |

represent values of s and c useful for computing the bulge factor of the arc.

Arc from Bounding Points and Exit Direction. [arcexit.html]

Let and be the start and end points of an arc, respectively, and P be a third point on the vector tangent to the arc at . Show that

s | = | |

c | = |

represent values of s and c useful for computing the bulge factor of the arc.

Midpoint of Arc. [arcmidpt.html]

Show that the midpoint, P of a bulge factor arc between points and whose bulge factor is B has coordinates

Centroid of Semicircular Arc. [arccent.html]

Show that the centroid of the area bounded by a semicircular arc of radius r and its chord is on the axis of symmetry at a distance

from the chord of the arc.

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