Arc from Bounding Points and Exit Direction

arcexit.html

Exploration

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

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

Approach

Use the trigonometric definition of a cross-product to justify the value for s. Use the trigonometric definition of a dot product to justify the value for c.

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`

Solution

Let Q be the point of intersection of the tangents at end points and .  The entry angle is equal to the angle because triangle is an isosceles triangle.  The cross-product definition in two dimensions is given by A×B=|A| |B|sin (α) where α is the angle between vectors A and B. Therefore, the expression is which is a scalar multiple of sin (α).  The dot product trigonometric definition in two dimensions is given by A·B=|A| |B|cos (α) where α is the angle between vectors A and B.   is and, therefore, is the same scalar multiple of cos (α).  Therefore, s and c are multiples of the sine and cosine of the angle between the chord and the entry angle as required.

Discussion

Example:  Construct and sketch the arc with end points (3,0) and (0,0) with an exit angle vector through the point (1,-1).  First define functions for the two-dimensional cross-product and magnitude.

Cross2D[{u1_,v1_},{u2_,v2_}]:=Cross[{u1,v1,0},{u2,v2,0}];

Magnitude2D[{u1_,v1_,w1_:0}]:=Sqrt[u1^2+v1^2+w1^2];

Compute the bulge factor using s and c.  The bulge factor is given by

.

P0=Point2D[p0={3,0}];
P1=Point2D[p1={0,0}];
P=Point2D[p={1,-1}];
s=Magnitude2D[Cross2D[p1-p0,p-p1]];
c=Dot[p1-p0,p-p1];
B=s/(c+Sqrt[c^2+s^2])

Plot the geometry.

Sketch2D[{P0,P1,P,Arc2D[p0,p1,B],Segment2D[P,P1]}]

Graphics saved as "arcexi01.eps".