Conjugation of a line with a circle

Practical work №2

Engineering graphics

“Theoretical part”

PAIRINGS

In the outlines of technical forms, there are often smooth transitions from one line to another. A smooth transition from one line to another, made using an intermediate line, is called conjugation . The construction of mates is based on the following positions of the geometry.

1. The transition of a circle into a straight line will be smooth only when the given straight line is tangent to the circle (Fig. 1a). The radius of the circle, drawn to the point of contact K , is perpendicular to the tangent line.

2. The transition from one circle to another at point K will only be smooth when the circles have a common tangent at a given point (Fig. 1b ).

Rice. one

The tangent point K and the centers of the circles O 1 and O 2 lie on the same straight line. If the centers of the circles lie on opposite sides of the tangent t , then the tangency is called external (Fig. 1 b ); if the centers O 1 and O 2 are on the same side of the common tangent – internal, respectively (Fig. 11 c ). In the theory of conjugations, the following terms are used: a ) the center of conjugation is the point O (Fig. 2); b ) junction radius R (Fig. 2); c ) conjugation points A and B (Fig. 2); d ) conjugation arc AB (Fig. 2).

The center of conjugation O is a point equidistant from the conjugated lines (Fig. 2).

The junction point A (B) is the point of contact of two mating lines (Fig. 2).

The conjugation arc AB is the arc of a circle with which conjugation is performed (Fig. 12).

The mate radius R is the radius of the mate arc (Fig. 12).

To perform conjugations, it is necessary to define three construction elements: 1) conjugation radius; 2) interface center; 3) junction points.

Pairing two intersecting straight lines

Let two intersecting straight lines m, n and the mate radius R be given (Fig. 2). It is necessary to construct a conjugation of these straight lines by an arc of a circle with radius R.

Rice. 2

Let’s make the following constructions.

1. Construct a set of points of conjugation centers remote from the straight line n at a distance of radius R of conjugation. Such a set is the line n / parallel to the given line n and at a distance R from it.

2. Let’s construct a set of points of conjugation centers remote from the straight line m at the distance of the conjugation radius. Such a set is the line m / , parallel to m and separated from the latter by a distance R .

3. At the intersection of the constructed lines m / and n / we find the conjugation center O .

4. Define the point A of conjugation on the line n . To do this, let’s drop the perpendicular from the center O to the line n. To determine the conjugation point B on the line m , it is necessary to lower the perpendicular from the center O to the line m , respectively. Draw a conjugation arc AB . Now all mate elements will be defined: radius, center and mate points.

Conjugation of a line with a circle

The conjugation of a straight line with a circle can be external or internal. Consider the construction of external conjugation of a straight line with a circle.

Example 1. Let a circle with radius R centered at the point O 1 and a straight line m be given. It is required to construct a conjugation of a circle with a straight arc of a circle of a given radius R (Fig. 3).

To solve the problem, we perform the following constructions.

1. Let’s construct a set of points of conjugation centers remote from the conjugated straight line at a distance R. This set defines a straight line m / , parallel to m and separated from it by a distance R .

2. The set of points of conjugation centers, remote from the circle n at a distance R, is a circle n / , drawn with a radius R 1 + R .

3. We find the conjugation center O as the point of intersection of the lines n / and m / .

4. We find the conjugation point A as the base of the perpendicular drawn from the point O to the line m . To build a conjugation point B, it is necessary to draw a line of centers OO 1 , i.e. connect the centers of conjugated arcs. At the intersection of the line of centers with a given circle, we define point B.

5. Let’s draw a conjugation arc AB .

Rice. 3 Fig. 4

Example 2. When constructing an internal conjugation (Fig. 4), the sequence of constructions remains the same as in example 1. However, the conjugation center is determined using an auxiliary circular arc drawn from the center O 1 with a radius R – R 1 .

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