1. Technical Drawing in Photonics
Lesson 4
Tangents.
Drawing of a parabola.
Drawing of a hyperbola.
Dr. Zsolt István Benkő
TAMOP C-12/1/KONV project
„Preparation of the concerned sectors for educational and R&D activities related to the Hungarian ELI project”

2. Tangent to a circle from a given external point
Lesson 4
Tangent to a circle from a given external point
There is a circle and external point. Connect the center of the circle and the external point and bisect the segment.
Technical drawing in Photonics

3. Tangent to a circle from a given external point
Lesson 4
Tangent to a circle from a given external point
Draw a Thales circle using the segment as a diameter. From Thales’ Theorem it is known that the endpoints of a diameter and an arbitrary third point on the circle form a triangle which has a right angle (90o) at the third point. It is also known that a tangent and the corresponding radius are perpendicular.
Technical drawing in Photonics

4. Tangent to a circle from a given external point
Lesson 4
Tangent to a circle from a given external point
The lines drawn from the external point through the intersections of the circles are tangents.
Technical drawing in Photonics

5. Tangent to a circle from a given external point
Lesson 4
Tangent to a circle from a given external point
Technical drawing in Photonics

6. Common external tangent to two circles
Lesson 4
Common external tangent to two circles
Connect the centers of the given circles and bisect the segment.
Technical drawing in Photonics

7. Common external tangent to two circles
Lesson 4
Common external tangent to two circles
Draw a concentric support circle inside the LARGER circle with a radius which is the difference of the radii of the original circles.
Technical drawing in Photonics

8. Common external tangent to two circles
Lesson 4
Common external tangent to two circles
Draw the tangents to the support circle from the center of the SMALLER circle.
Technical drawing in Photonics

9. Common external tangent to two circles
Lesson 4
Common external tangent to two circles
Draw the corresponding radii in the LARGER circle.
Technical drawing in Photonics

10. Common external tangent to two circles
Lesson 4
Common external tangent to two circles
Using these radii slide the tangents parallelly to the original circles.
Technical drawing in Photonics

11. Common external tangent to two circles
Lesson 4
Common external tangent to two circles
Technical drawing in Photonics

12. Common internal tangent to two circles
Lesson 4
Common internal tangent to two circles
Connect the centers of the given circles and bisect the segment.
Technical drawing in Photonics

13. Common internal tangent to two circles
Lesson 4
Common internal tangent to two circles
Draw a concentric support circle outside of EITHER circle with a radius which is the sum of the radii of the original circles.
Technical drawing in Photonics

14. Common internal tangent to two circles
Lesson 4
Common internal tangent to two circles
Draw the tangents to the support circle from the center of the OTHER circle.
Technical drawing in Photonics

15. Common internal tangent to two circles
Lesson 4
Common internal tangent to two circles
Draw the corresponding radii in the FIRST circle.
Technical drawing in Photonics

16. Common internal tangent to two circles
Lesson 4
Common internal tangent to two circles
Using these radii slide the tangents parallelly to the original circles.
Technical drawing in Photonics

17. Common internal tangent to two circles
Lesson 4
Common internal tangent to two circles
Technical drawing in Photonics

18. Common external touching circle to two circles
Lesson 4
Common external touching circle to two circles R
Two circles and the R radius of the touching circle are given. R must be larger than the half of the distance between the two circles.
Draw arcs from each center with a radius which is a sum of R and the radius of the same circle.
Technical drawing in Photonics

19. Common external touching circle to two circles
Lesson 4
Common external touching circle to two circles
These intersecting arcs are the centers of the touching circles.
Technical drawing in Photonics

20. Common internal touching circle to two circles
Lesson 4
Common internal touching circle to two circles R
Two circles and the R radius of the touching circle are given. R must be larger than the half of the distance between the most distant points of the two circles.
Draw arcs from each center with a radius which is the difference of R and the radius of the same circle.
Technical drawing in Photonics

21. Common internal touching circle to two circles
Lesson 4
Common internal touching circle to two circles
These intersecting arcs are the centers of the touching circles.
Technical drawing in Photonics

22. Drawing of a parabola (parabolic curve)
Lesson 4
Drawing of a parabola (parabolic curve) d F
The points of a parabola are equally distant from a line (directrix: d) and a point (focus: F).
Draw a normal line from the focus to the directrix and bisect it. The bisection point is the vertex of the parabola and this line is the symmetry axis. The drawn segment is the ”p” parameter of the parabola.
Technical drawing in Photonics

23. Drawing of a parabola (parabolic curve)
Lesson 4
Drawing of a parabola (parabolic curve) d F
Draw arbitrary parallel lines with the directrix above the vertex.
Technical drawing in Photonics

24. Drawing of a parabola (parabolic curve)
Lesson 4
Drawing of a parabola (parabolic curve) d F
Set the pair of compasses to the distance of a chosen line from the directrix and draw arcs to the same line from the focus.
Technical drawing in Photonics

25. Drawing of a parabola (parabolic curve)
Lesson 4
Drawing of a parabola (parabolic curve) d F
These points denote the parabola. Connect them by a smooth line.
Technical drawing in Photonics

26. Drawing of a hyperbola (hyperbolic curve)
Lesson 4
Drawing of a hyperbola (hyperbolic curve) a c F1 F2 C
The hyperbola (hyperbolic curve) has two focal points (F1, F2). The half of the major axis is ”a”. The half of the distance of the focal points is ”c”. ”c” has to be larger than ”a”. ”C” is the center of the hyperbola, the bisecting point between the two focal points.
The hyperbola has two distinct curves.
The points of the curves are defined by this way: the difference of the distances of a certain point on the curve from F1 and from F2 equals to 2a.
Technical drawing in Photonics

27. Drawing of a hyperbola (hyperbolic curve)
Lesson 4
Drawing of a hyperbola (hyperbolic curve) a c F1 F2 C
A rectangle should be drawn with sizes of 2a x 2c centered to C. The intersections of the rectangle and the major axis are the vertices of the hyperbola; they are the closest points of the two curves.
Technical drawing in Photonics

28. Drawing of a hyperbola (hyperbolic curve)
Lesson 4
Drawing of a hyperbola (hyperbolic curve) a c F1 F2 C
The extended diagonals of the rectangle are the asymptotes of the hyperbola. The curves go to infinity and are closer and closer to the asymptotes but never touching them.
Technical drawing in Photonics

29. Drawing of a hyperbola (hyperbolic curve)
Lesson 4
Drawing of a hyperbola (hyperbolic curve) a c F1 F2 C
With an arbitrary radius which is larger than c + a, arcs should be drawn from the focal points to the direction of the other focal point. By subtracting 2a from the radius new arcs should be drawn to intersect with the previously drawn arcs.
Technical drawing in Photonics

30. Drawing of a hyperbola (hyperbolic curve)
Lesson 4
Drawing of a hyperbola (hyperbolic curve) a c F1 F2 C
These intersections are the points of the two symmetric curves. They should be connected by a smooth line.
Technical drawing in Photonics

31. Lesson 4
References
Ocskó Gy., Seres F.: Gépipari szakrajz, Skandi-Wald Könyvkiadó, Budapest, 2004
Lőrincz P., Petrich G.: Ábrázoló geometria, Nemzeti Tankönyvkiadó Rt., Budapest, 1998
Pintér M.: AutoCAD tankönyv és példatár, ComputerBooks, Budapest, 2006
Technical drawing in Photonics