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19. Conjugation of two arcs of circles by the third arc

Lecture



When constructing the conjugation of two arcs of circles by the third arc of a given radius, we can consider three cases: when a conjugating arc of radius R touches given arcs of radii R 1 and R 2 from the outside (Fig. 34, a); when it creates an internal touch (fig. 34, b); when internal and external tangencies are combined (Fig. 34, c).

The construction of the center O of the conjugating arc of radius R with an external touch is carried out in the following order: from the center O 1 with a radius equal to R + R 1 , an auxiliary arc is drawn, and from the center O 2 an auxiliary arc with radius R + R 2 is conducted . At the intersection of the arcs, a center O of a conjugate arc of radius R is obtained , and at the intersection of radius R + R 1 and R + R 2 with arcs of circles, there are obtained conjugation points A and A 1 .

The construction of the center O with internal tangency differs in that from the center O 1 they carry out an auxiliary circle with a radius equal to R - R 1 and from the center O 2 with a radius R - R 2 . When combining internal and external tangency from the center O 1 , an auxiliary circle with a radius equal to R - R 1 and from the center O 2 is carried out with a radius equal to R + R 2 .

How to build internal and external mates of a given radius for circles?
 
When constructing the conjugation of two circles by the arc of the third circle of a given radius, three options are possible: external conjugation, internal conjugation, and a combination of external and internal conjugations.
 
Before building mates in AutoCAD, let's remember what mating is and how mates are built for two circles.
 
Conjugation is the smooth transition of one line (straight or curved) to another. The point at which one line passes to another is called a pairing point . When drawing mates, it is necessary to construct the center of the conjugating arc and determine the points of conjugation or tangency.
 
Let us analyze in order each variant of the conjugation of two circles.
 
 
External conjugation of circles by an arc of a given radius R.
 
The mating arc touches specified circles on the outside. The center O of the conjugating arc must be separated from the circles at the same distance, equal to R. To construct the center O of the conjugating conjugating arc, from the centers of the circles O 1 and O 2, we draw two auxiliary arcs with radii ( R + R 1 ) and ( R + R 2 ) until they intersect each other. The points of conjugation lie on the lines connecting the centers of the circles.

19. Conjugation of two arcs of circles by the third arc

Answers to users' questions on AutoCAD

Copyright 2007-2012 Ivan Boyarsky.

19. Conjugation of two arcs of circles by the third arc

Well, I hope everything is clear with theory. We proceed to the implementation of the foregoing with respect to AutoCAD. Of course, you can go to the specified path, build auxiliary circles to find the center of the arc of a pairing, build a mate and remove auxiliary circles. You can go more simple way. AutoCAD has the FILLET tool, but it can only build an external mate :(
 
How to deal with the other two cases of pairings?
 
There is an exit! And it is quite simple! As they say, all ingenious is simple!
 
To build all the specified mates, it is enough to construct a circle at two points of tangency with a given radius of the mate arc! In the above drawings, the touch points are indicated by small circles. Well, then only remains to trim the excess ...
 
How to build a circle at two points of contact with a given radius?
 
To do this, go to the menu Drawing - Circle :

Internally, the conjugation of circles by an arc of a given radius R.
 
The mating arc touches specified circles by the inner side. The center O of the conjugating arc is determined by the intersection of the arcs of the auxiliary circles, the radii of which are equal to the differences ( RR 1 ) and ( RR 2 ).

19. Conjugation of two arcs of circles by the third arc

The combination of external and internal conjugations of circles by an arc of a given radius R.
 
One of the given circles is inside the mating circle. The center O of the conjugating arc is determined at the intersection point of the auxiliary circles drawn for the external conjugation with a radius ( R + R 1 ), and for the inner circumference - with a radius ( RR 2 ).

19. Conjugation of two arcs of circles by the third arc

I hope you have succeeded.

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3 Some geometric constructions

Terms: 3 Some geometric constructions