This document provides an overview of conic sections through a presentation. It begins by introducing the current curriculum objectives. It then discusses examples of conic sections found in nature and defines different types of cones. The main types of conic sections - circle, ellipse, parabola, and hyperbola - are examined based on the angle between the cutting plane and the cone's axis. Their common features like vertices, foci, axes, and directrices are also outlined. The document demonstrates conic sections in GeoGebra and provides nets for building solid models. Finally, it briefly traces the historic origins and early studies of conic sections in ancient Greece.

1. Conic Sections © Bijendra Bir Karmacharya 2020
Conic Sections
[ Day 1 ]
Presented by:
Bijendra Bir Karmacharya
Organized by:
Mathematics Connection and Research in Nepal
July 26-30, 2020

2. Conic Sections © Bijendra Bir Karmacharya 2020
Contents of this presentation
 Current Curriculum and objectives
 Curves and sections in our nature
 The Cone
 Parts of Right Circular Cone
 Single and double napped cones
 Conic Sections
 Angle between line and plane
 Analyzing the cases in terms of Axis-plane angle
 Comminalities of conic sections
 Demonstration in GeoGebra
 Nets for making solid samples
 Historic background
 Literal meanings of Ellipse, Parabola, and
Hyperbola
 Sample Project Works
 Sample questions for evaluation (K & U)

3. Conic Sections © Bijendra Bir Karmacharya 2020
Current Curriculum and objectives

4. Conic Sections © Bijendra Bir Karmacharya 2020
Curves and sections in our nature
It is always better to start new concepts from Concrete -- Semi-Abstract -- Abstract sequence. In the
context of Conic sections, we can show plenty of daily life examples of circular, cylindrical and conic section
and slowly relate them into four conic shapes and finally introduce the equation as abstract concepts.
Concrete
Examples
[spherical, cylindrical
and conic cuts]
Semi-Abstract
Concept
[figures of conic
sections]
Abstract
Conceptualization
[Equations]

5. Conic Sections © Bijendra Bir Karmacharya 2020
Various Circular cuts
Discussing different sections of sphere and cylinders
Convincing them every spherical section are circles.
What shapes are there? What type of sections are formed?

6. Conic Sections © Bijendra Bir Karmacharya 2020
Various elliptical cuts
Is it possible to cut elliptical from sphere?
What shapes are possible from cylindrical sections?

7. Conic Sections © Bijendra Bir Karmacharya 2020
Various conical shapes
Can we cut circular section in cone?
Can we cut elliptical section in cone?
Is it possible to get other type of sections?
What type of cones are possible?

8. Conic Sections © Bijendra Bir Karmacharya 2020
The Cone
Mostly we can see the diagrams depicted in Right Circular Cone, but
theoretically Cones are defined much broader.
"A cone is a surface generated by a straight line which passes through
a fixed point (called vertex) and intersects a given curve."
Robert J. T. Bell (1910) An elementary Treatise on Coordinate
Geometry of Three Dimensions. Macmillan India Ltd. 3e.
Arbitrary curve based
Cone

9. Conic Sections © Bijendra Bir Karmacharya 2020
The Cone (contd.)
Our special attention goes to circular cones. When the curve is circle,
the cone is called Circular cone.
The line joining Vertex and the centre of the circle is called Axis of the
Circular Cone.
When the Axis is perpendicular to the plane of circle, the cone is called
Right Circular Cone. And, when the Axis is not perpendicular to the
plane of circle, it is called Oblique Circular Cone. Here we take Right
Circular Cone as the Standard Cone.
Right Circular Cone
Oblique Circular Cone

10. Conic Sections © Bijendra Bir Karmacharya 2020
Parts of Right Circular Cone
Semi-Vertical Angle is
uniform or constant in Right
Circular Cone. But it is not
uniform in Oblique Circular
Cone.
Vertex
Semi-vertical Angle
Axis line
Height
Base Circle
Base Radius
Generator line

11. Conic Sections © Bijendra Bir Karmacharya 2020
Single and Double Napped cones
Double Napped Cone
Single Napped Cone

12. Conic Sections © Bijendra Bir Karmacharya 2020
Conic Sections
When a plane cuts a Right Circular Cone, there are multiple cases of
intersections.
Degenerate cases: Plane passes thru vertex.
Degenerate Conics
 Cutting through vertex only  A single point
 Tangent to the cone  A single line
 Cutting the cone but passing through the vertex
 A pair of line
श ांकिि क्षेत्रहरू
सोलीि िलमीहरू
शांिू च्छेदहरू
Point Line Line Pair

13. Conic Sections © Bijendra Bir Karmacharya 2020
Conic Sections (contd.)
Non-Degenerate Cases : Plane does not pass thru the vertex.
Conic Sections
 Circle
 Ellipse
 Parabola
 Hyperbola
What makes these all cases so different?

14. Conic Sections © Bijendra Bir Karmacharya 2020
Angle between the Line and Plane
In 3D, angle between plane and a line is defined as the complementary
of the angle made by the line with the normal to the plane. For
non-directional use, angle between plane and line is supposed to vary
from 0° to 90° only.
Angle between plane and angle = 90° – 

line
plane
normal

15. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-plane angle
Let the semi-vertical angle = 
Axis-Plane angle = [angle supposed to vary from 0° to 90°]
Vertex = V
Degenerate Conic : Single Point
Plane Passing through V = Yes
Axis-Plane angle  θ > α

16. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-plane angle (contd.)
Let the semi-vertical angle = 
Axis-Plane angle = [angle supposed to vary from 0° to 90°]
Vertex = V
Degenerate conic : Single Line
Plane Passing through V = Yes
Axis-Plane angle  θ = α

17. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-plane angle (contd.)
Let the semi-vertical angle = 
Axis-Plane angle = 
Vertex = V
Degenerate conic : Line Pair
Plane Passing through V = Yes
Axis-Plane angle  θ < α

18. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-plane angle (contd.)
Summarizing Degenerate conics in terms of Axis-Plane angle in a
number line:
θ
α
Pair of lines
0° ≤ θ < α
0° 90°
Single line
θ = α
Single Point
α < θ ≤ 90°

19. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-Plane angle (contd.)
Let the semi-vertical angle = 
Axis-Plane angle = [angle supposed to vary from 0° to 90°]
Vertex = V
Conic Section : Circle
Plane Passing through V = No
Axis-Plane angle  θ = 90°

20. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-Plane angle (contd.)
Let the semi-vertical angle = 
Axis-Plane angle = [angle supposed to vary from 0° to 90°]
Vertex = V
Conic Section : Ellipse
Plane Passing through V = No
Axis-Plane angle  α < θ < 90°

21. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-Plane angle (contd.)
Let the semi-vertical angle = 
Axis-Plane angle = [angle supposed to vary from 0° to 90°]
Vertex = V
Conic Section : Parabola
Plane Passing through V = No
Axis-Plane angle  θ = α

22. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-Plane angle (contd.)
Let the semi-vertical angle = 
Axis-Plane angle = [angle supposed to vary from 0° to 90°]
Vertex = V
Conic Section : Hyperbola
Plane Passing through V = No
Axis-Plane angle  θ < α

23. Conic Sections © Bijendra Bir Karmacharya 2020
Analyzing the cases in terms of Axis-plane angle (contd.)
Summarizing Conic sections in terms of Axis-Plane angle in a
number line:
Circle
θ = 90°
θα
Hyperbola
0° ≤ θ < α
0° 90°
Parabola
θ = α
Ellipse
α < θ < 90°

24. Conic Sections © Bijendra Bir Karmacharya 2020
Comminalities of conic sections
Though conic sections are defined in 3D background, studied in 3D
originally, they are 2D figures. They all have some commonalities:
Vertex: It is the turning point of the conic section. Circle being perfectly
symmetric, no special vertex is assigned to it. Denoted here as A's.
Focus: It is a point of analytic importance. All conic sections have two
foci, though circle has overlapping focus called as centre. They are
mostly denoted by S.

25. Conic Sections © Bijendra Bir Karmacharya 2020
Comminalities of conic sections (contd.)
Axis: A line passing through Vertex and focus of a conic section is called
Axis of the conic section. In case of circle as there is no special vertex,
there is no specially assigned axis also. [Red lines in figures]
Directrix: It is a special line associated with the conic section. It is
perpendicular to the axis of the conic and is positioned such that, for
any point on the conic, ratio of its distance from focus and
perpendicular distance to this line is always constant. In case of
circle, directrix is supposed to be at infinity. [Green Lines in figures]
Eccentricity: The constant ratio mentioned above is called eccentricity.

26. Conic Sections © Bijendra Bir Karmacharya 2020
Demonstration in GeoGebra

27. Conic Sections © Bijendra Bir Karmacharya 2020
Nets for making solid samples
Some nets for cut frustums are developed here.
To make the solid model, please print and cut the shape.
While cutting, DO NOT FORGET TO KEEP THE FOLDING
PART OUTSIDE THE CREASE for pasting purpose outside the printed
border.
© Bijendra 2020
Semi-Vertical angle = 30°
Axis-Plane angle = 30°

28. Conic Sections © Bijendra Bir Karmacharya 2020
© Bijendra 2020 © Bijendra 2020

29. Conic Sections © Bijendra Bir Karmacharya 2020
© Bijendra 2020

30. Conic Sections © Bijendra Bir Karmacharya 2020
Historic background
The knowledge of conic sections can be traced back to Ancient Greece.
Menaechmus is credited with the discovery of conic sections around the
years 360-350 B.C.
He used them in his two solutions to the Delian problem or "doubling
the cube" problem.
Side =
3
2 units
Volume = 2 c.u.
Side = 1 unit
Volume = 1 c.u.

31. Conic Sections © Bijendra Bir Karmacharya 2020
Historic background (contd.)
Aristaeus and Euclid investigated these curves after Menaechmus.
Euclid

32. Conic Sections © Bijendra Bir Karmacharya 2020
Historic background (contd.)
Archimedes made major contributions to the growth of conic section
But it does not appear that he published any work devoted solely to them.
Archimedes

33. Conic Sections © Bijendra Bir Karmacharya 2020
Historic background (contd.)
In late 3rd – early 2nd centuries BCE, Apollonius, wrote an eight-book
series on the Conic Sections.
He is now known as the "Great Geometer" on the basis of this text.
The first four books have been well preserved in the original Ancient
Greek.
Books V-VII are known only from its Arabic translation.
The eighth book has been lost entirely.
Apollonius

34. Conic Sections © Bijendra Bir Karmacharya 2020
Literal meanings of Ellipse, Parabola, and Hyperbola
The word Ellipse (इलिप्स्) came out of the ancient Greek word
ἐλλείπω (elleípō, इलिपो) which means "to fall short", कम देलिनु)
the word ellipsis ("…") (इलिप्सीस्) also came from same word (used for
shortening the text, meaning omission of writing for understood context
or out of context words.)(सन्दभभ अनुसार स्वभालवक रूपमा बुझ्न सलकने, वा
असान्दलभभक भएका शब्दहरू निेलि लेख इ छोट्य उनिो िालि प्रयोि िररने तीन थोप्िे
लिह्न "…") । नेपािीमा Ellipse िाई दीर्घवृत्त भलनन्छ ।

35. Conic Sections © Bijendra Bir Karmacharya 2020
Literal meanings of Ellipse, Parabola, and Hyperbola (contd.)
The word Parabola (प र बोल ) came out of the ancient Greek word
παραβάλλω (parabállō, पाराबाल्िो), which means "set side by
side” (संि संिै जानु, बराबरी जानु). The word Parallel also came from the
same word, meaning same distanced, or set side by side (समान दुरीमा,
समान रूपिे)

36. Conic Sections © Bijendra Bir Karmacharya 2020
Literal meanings of Ellipse, Parabola, and Hyperbola (contd.)
The word Hyperbola came out of the ancient Greek word ὑπερβολή
(huperbolḗ)(इपरबोि) which means "to be excess", (बलि हुनु). The word
hyperbole (ह इपःबोली) also came from the same word, which means
exaggeration or overstatement. (बिाइ ििाइ िरेर बोलिएको)
.

37. Conic Sections © Bijendra Bir Karmacharya 2020
Sample Project Works
There are so many daily life instances where we can deal with conic
sections. Some of the easy ways to show pupil are listed below.
Collect some cylindrical objects (like bamboo or pipe pieces) and cut
them perpendicularly and in oblique angles. What shapes do you observe?
केही बेिनाकार बस्तुहरू (बााँस वा पाइपका टुक्राहरू) जम्मा पानुभहोस् र त्यसिाई िम्ब र टेिो
िरी काट्नुहोस् । यसरी काट्दा कस्ता आकृलत देिा पछभन् िेख्नुहोस् ।

38. Conic Sections © Bijendra Bir Karmacharya 2020
Sample Project Works (contd.)
A glass of conical frustum shape is half filled with water. Can you show
different conic sections using the level surface of water? (This can be
done by using conical flask from science lab.) एउटा फ्रस्टम (कालटएको सोिी)
आकारको लििासमा आधा पानी रालिएको छ । के पानीको सतहबाट लवलभन्न कोलनक
सेक्सनहरू देिाउन सलकन्छ ? (यसिाई लवज्ञान प्रयोिशािाको कोलनकि फ्िास्क प्रयोि िरर
पलन देिाउन सलकन्छ ।)

39. Conic Sections © Bijendra Bir Karmacharya 2020
Sample Project Works (contd.)
Prepare four conical wax shapes and demonstrate four conic sections
cutting them in different angles. (Soaps can be used alternatively instead
of wax.) िारवटा मैनका सोिी आकारहरू बनाउनुहोस् र त्यसिाई फरक फरक कोणमा
काटेर िार लकलसमका कोलनक सेक्सन प्रस्तुत िनुभहोस् । (मैनको सट्टा साबुन पलन प्रयोि िनभ
सलकन्छ । )

40. Conic Sections © Bijendra Bir Karmacharya 2020
Sample questions for evaluation (K & U)
 What shape is formed when a right circular cone is cut parallel to the base? िम्ब
वृत्ताकार सोिीिाई आधार संि समानान्तर हुने िरी काट्दा कस्तो आकृलत देलिन्छ ?
 In which angle a right circular cone is to be cut by a plane to form a parabola?
पाराबोिा आकृलत बनाउन एउटा िम्ब वृत्ताकार सोिीिाई एउटा समति सतहिे कुन कोणमा काट्नु
पछभ होिा ?
 A plane cuts a cone making 45° to the axis of the right circular cone. If the semi-
vertical angle of the cone is 35°, which conic section is formed there? एउटा
समति सतहिे एउटा िम्ब वृत्ताकार सोिीिाई यसको अक्ष संि 45° बनाएर काटेको छ । यलद सो
सोिीको शीषाभधभ कोण 35° छ भने कस्तो कोलनक सेक्सनको आकृलत बनेको होिा ?

41. Conic Sections © Bijendra Bir Karmacharya 2020
Thank You
9 8 4 1 2 7 1 5 6 7
https://www.facebook.com/People.cadres
veejayandra@gmail.com