FOUR CONIC SECTIONS. Sections of a Cone

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Samuel Newman
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1 Conic Sections

2 FOUR CONIC SECTIONS 1 Sections of a Cone

3 The circle, ellipse, parabola and hyperbola are known as conic sections Circle Ellipse Parabola Hyperbola All four curves are obtained by slicing a double cone at varying angles

hyperbola is an ellipse Generators are lines we put on the surface of curved objects Ellipse Circle Parabola Hyperbola The hyperbola The

4 When When a When plane a plane a this cuts plane cuts plane all is CONICS generators both parallel is perpendicular parts AS to of on one SECTIONS the one generator, double side the of cone, OF axis, the the A apex, the CONE section section the is section a is parabola a circle hyperbola is an ellipse Generators are lines we put on the surface of curved objects Ellipse Circle Parabola Hyperbola The hyperbola The parabola The has ellipse two has branches and only circle one extending branch are closed and indefinitely curves is unlimited in opposite directions

5 THE ELLIPSE AROUND US The circle is the simplest of all curves However, the ellipse is the curve most often "seen" in everyday life This is because every circle, viewed obliquely, appears elliptical The orbits of the planets and satellites are ellipses

6 The ellipse is without doubt one of the most popular curves to be seen. For example, every time you look at a circular figure at an oblique angle (one that is not a right angle), the curve you see is an ellipse. In the example shown, when you look in perpendicular to the piece you can see that it is circular but as it rotates you see that it becomes elliptical, this is also evident when looking in at the top of your cup of tea or when looking at the top of any cylindrical object

7 Orthographic Projection + Developments Part 2: Handout 7 We have already drawn an ellipse and we did not know it!! There are a number of ways to draw the ellipse and not the rest of the object

8 This is not the True shape of the Ellipse. When we are drawing an ellipse using different methods, we will always be finding the True Shape of the Ellipse

9 WHISPERING GALLERIES In rooms where the ceilings are elliptical, a sound made at one focus can be heard very clearly at the other focus. Any sound made at the focus will rebound of the elliptical ceiling and pass through the other focus.

When a ball passes over the other focal point, regardless of which point on the ellipse it

10 ELLIPTICAL BILLIARDS TABLE The elliptical billiards table uses the same principle as the whispering gallery above. The pocket is positioned at one of the focal points. When a ball passes over the other focal point, regardless of which point on the ellipse it bounces off, it will always pass over the other focal point. Shown below is a sample of a path traced by a ball on the table.

11 Ellipses can be seen extensively in logos. Here is one example of a logo used by ford motors which is in the shape of an ellipse. What make of car is this? What sports game uses a 3-D Ellipse?

12 THE ELLIPSE AND THE CYLINDER As can be seen, when a cylinder is cut by a plane the true shape of the section is an ellipse To find the focal points of the curve the focal spheres can be used as were used with the cone.

13 METHODS OF CONSTRUCTING AN ELLIPSE There are numerous methods of constructing ellipses, here we will take a look at the more popular methods. 1. AUXILIARY CIRCLES METHOD 2. THE PIN AND STRING METHOD 3. THE TRAMMEL METHOD 4. AN ELLIPSE IN A RECTANGLE

14 1. AUXILIARY CIRCLES METHOD PROCEDURE 1. Draw the major and minor axis. 2. Construct the major and minor auxiliary circles as shown. 3. Divide the circles into a number of different segments.(using 30 degree set-square as shown) 4. Where the line intersects the minor auxiliary circle project a line across parallel to the major axis and where the line intersects the major auxiliary circle project a line parallel to the minor axis until where it meets the previous line drawn. 5. Complete the construction for the rest of the points and draw in the curve.

18 2. THE PIN AND STRING METHOD This method of drawing an ellipse has been used for hundreds of years. You can try it by sticking two thumbtacks in a sheet of paper and hooking a piece of string around them. Keep the string stretched with the point of your pencil and move your pencil around to trace an ellipse. PRINCIPLE The two thumbtacks represent the foci and since the length of the string does not change it proves that the sum of the distances (PF) and (PF1) [P= point on curve, F= foci] is a constant and this constant is the major axis.

19 3. TO CONSTRUCT AN ELLIPSE IN A RECTANGLE PROCEDURE 1. Divide the sections of the rectangle into a number of equal parts as shown. 2. Draw a line from the sub-sections on the sides of the rectangle to the top of the minor axis. 3. Project a line from the bottom of the minor axis to meet these lines to locate points on curve. 4. Axial symmetry can be used to locate the other half of the curve

20 minor axis 5 equal divisions ELLIPSE major axis 5 equal divisions The same number of equal divisions must be used for each quadrant

21 4. THE TRAMMEL METHOD This is also a very old method of constructing an ellipse, it was Archimedes who first used this method. You can try it by cutting out a strip of paper/cardboard and marking it as shown; PROCEDURE 1. Position the trammel so that the point B lies on the major axis and the point C lies on the minor axis, mark the location of point A - this is a point on the curve. 2. Rotate the trammel keeping B on the major axis and C on the minor axis, mark the position of a all the time 3. Join the various points and the resulting curve is an ellipse.

22 THE TRAMMEL METHOD

23 THE PARABOLA AROUND US When a sliothar or golf ball is hit into the air, it follows a parabolic path The centre of gravity of a leaping salmon describes a parabola The parabola is used by engineers in designing some suspension bridges

THE PARABOLA AROUND US If a light is placed at the focus of a parabolic mirror (a curved surface formed by rotating a parabola about its axis), the light will be reflected in rays parallel to the

24 THE PARABOLA AROUND US If a light is placed at the focus of a parabolic mirror (a curved surface formed by rotating a parabola about its axis), the light will be reflected in rays parallel to the axis This property is used in the design of flashlights and headlights The bulb is placed at the focus for the high beam and a little above the focus for the low beam The opposite principle is used in the giant mirrors in reflecting telescopes used to collect light and radio waves from outer space The beam comes toward the parabolic surface and is brought into focus at the focal point

25 4 equal divisions axis PARABOLA vertex The same number of equal divisions must be used for each half 4 equal divisions 1

26 PARABOLA 1. Divide the sides of the rectangle into a number of equal points as shown. 2. Join the points on the side of the parabola to V (vertex) as shown. 3. Project lines parallel to the axis from the points on the bottom of the curve. 4. Where the corresponding lines intersect gives the required curve.

THE ELLIPSE AROUND US In an ellipse, any signal (light or sound) that starts at one focus is reflected to the other focus This principle is used in lithotripsy, a medical procedure for treating

27 THE ELLIPSE AROUND US In an ellipse, any signal (light or sound) that starts at one focus is reflected to the other focus This principle is used in lithotripsy, a medical procedure for treating kidney stones The patient is placed in a elliptical tank of water, with the kidney stone at one focus High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it The electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus

This early Greek study was largely concerned with the geometric properties of conics.

This early Greek study was largely concerned with the geometric properties of conics. 4.3. Conics Objectives Recognize the four basic conics: circle, ellipse, parabola, and hyperbola. Recognize, graph, and write equations of parabolas (vertex at origin). Recognize, graph, and write equations