The projections of a right cone are shown below. The traces of a simply inclined plane VTH are also given. The plane is parallel to an element of the cone. The intersection of a plane and a right cone is shown. The plane is parallel to an element of the cone. The line of intersection between the cone and the plane is shown in true size in the auxiliary view. (a) Determine the plan of the line of intersection between the cone and the plane. (a) Determine the position of the focal sphere in elevation. (b) Determine the true size of this line of intersection. (b) Establish the position of the focal point, latus rectum and directrix of the parabola in the auxiliary view. The curve is called a. horizontal cutting planes Conic Sections 1 NAME: DATE: 1
Work out the ratios for each of the points 1-5 Position 1 Position 2 Position 3 Position 4 Position 5 This ratio is called the eccentricity of the curve The eccentricity of a parabola is always A photograph of a Cleveland launcher driver is shown. The underside of the club is in the shape of a parabola. The drawing below shows the directrix, axis and focal point of a parabola. Determine the position of the vertex and draw a portion of the curve. Conic Sections 2 NAME: DATE: 2
The projections of a right cone are shown below. The traces of a simply inclined plane VTH are also given. The plane is placed so as to cut each and every element of the cone. (a) (b) Determine the plan and true size of the line of intersection between the cone and the plane. On the drawing on the right, determine the position of the focal sphere in elevation and establish the position of the focal points and directrices of the ellipse in the auxiliary view. horizontal cutting planes The curve is called an. Conic Sections 3 NAME: DATE: 3
Work out the ratios for each of the points 1-4 Position 1 Position 2 Position 3 Position 4 This ratio is called the eccentricity of the curve The eccentricity of an ellipse is always less than The drawing below shows the directrix, axis and focal point of an ellipse. The eccentricity of the curve is 3/4. Determine the position of the vertices, the minor axis and draw the curve. Conic Sections 4 NAME: DATE: 4
The drawing below shows a double hyperbola which is composed of the intersections of a cutting plane and both nappes of the double cone. On the drawing, determine the position of the focal spheres in elevation and establish the position of the focal points, directrices and asymptotes of the curve in the plan. When a pencil that has a hexagonal cross section is sharpened, a double hyperbola is formed. This results from the intersection of the conical point of the pencil by one of its flat sides. Conic Sections 5 NAME: DATE: 5
Work out the ratios for each of the points 1-4 Position 1 Position 2 Position 3 Position 4 This ratio is called the eccentricity of the curve The e of a hyperbola is always than 1 The drawing below shows the directrix, axis and focal point of a hyperbola. The eccentricity of the curve is 4/3. Determine the position of the vertex and draw a portion of the curve. Conic Sections 6 NAME: DATE: 6
A photograph of a Parabola Hall Table designed by Nathan Hunter is shown. Two curly wood panels intersect steel parabolas to support a glass top. The curved panels are portions of an ellipse. The incomplete elevation of the table is given below. The two parabolas are to be inscribed in the rectangles ABCD and EFGH, respectively. P is a point on the curve of the ellipse whose minor axis MN is drawn. Complete the view of the hall table showing all construction lines. P D C H G M N A B E F CONIC SECTIONS 7
The St. Louis Science Centre Planetarium's exterior curved surface is in the shape of a hyperboloid of revolution. This shape is formed by revolving a hyperbola around its axis. The drawing below shows an incomplete elevation of the planetarium. The line VV represents the transverse axis of the double hyperbola and P and Q are points on the curve. Complete the elevation by constructing the double hyperbola. V V P Q CONIC SECTIONS 8
When a plane goes faster than the speed of sound, a sonic boom shock wave occurs. A cone shaped wave shoots out of the back of the plane as shown in the photograph over and intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. D In the drawing over the line DD represents the directrix of a hyperbola and V is the vertex of the curve. The eccentricity is 1.25. Locate the focal point of the hyperbola and draw a portion of the curve. axis V CONIC SECTIONS 9 D
When a pencil that has a hexagonal cross section is sharpened, a hyperbola is formed. This results from the intersection of the conical point of the pencil by one of its flat sides. Two lines VF anf FD are shown in the drawing below. V is the vertex of a hyperbola, F is a focal point of the curve and D is a point on the directrix. Locate the directrix and eccentricity line and draw a portion of the curve. Key Principles: If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed. D V F CONIC SECTIONS 10
One of nature's best known approximations to parabolas is the path taken by a ball hit into the air. The friction of air and the pull of gravity will change the path slightly from that of a true parabola, but the error is insignificant in most cases. The drawing below shows the parabolic path of a golf ball as it bounces on the ground. All three parabolas are similar. Reproduce the drawing below showing all construction lines. Key Principles: The parabola is not a family of curves like the ellipse and hyperbola. The impression that some parabolas are more curved than others is because we are looking at different scales of the curve. 55 70 CONIC SECTIONS 11
If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed. Such an intersection can occur in physical situations as simple as the shadow formed on a wall by a lamp. Key Principles: The drawing below gives the location of the focal points of a double hyperbola as well as a point P on the curve. (a) Find the transverse axis and draw a portion of the double curve. (b) Draw the asymptotes to the curve. P F 1 2 F CONIC SECTIONS 12
The Gateshead Millenium Bridge spanning the river Tyne in England is a pedestrian and cycle bridge, instead of a stereotypical automobile bridge. The pedestrian/cycle pathway is a horizontal parabola, suspended above the river from a parabolic arch. The two parabolas are identical. The drawing below shows the incomplete elevation and plan of the bridge in the rest state as shown in the photograph across. (a) Draw the plan of the pathway by inscribing a parabola in the given rectangle. (b) Draw the plan of the parabolic arch. Include the projections of nine cables. Key Principles: The hyperbola may be defined as the locus of all points the difference of whose distances from two fixed points (focal points) is a constant. The asymptotes pass through the centre and are tangential to the curve at infinity. The whole bridge rotates as a single rigid structure as shown in the photograph below to allow boats to pass underneath. The geometry of the curves at each stage of the bridge's movement are superbly elegant. Shown below is the complete elevation and incomplete plan of the bridge after it is rotated through 20 degrees. Project a plan of the two parabolas in this position. CONIC SECTIONS 13
A photograph of a Cleveland Launcher driver is shown. The underside of the club is in the shape of a parabola. A drawing of this parabola is also shown. In the drawing below, F is the focus of the parabola and the line AFB represents a focal chord of the curve. (a) Determine the position of the directrix and the axis of the parabola. (b) Draw the portion of the curve as shown. A 90 F B CONIC SECTIONS 8
The Norman Foster designed London City Hall, known as "The Glass Egg", is a building that from the front looks like a boiled egg. Each level is offset slightly from the one below resulting in a slightly curving side and a rapidly curving side. In the drawing below, A and B are points on the curve of a parabola. The focus F is 95mm from A and 85mm from B. The centre of the circular arc BC lies on the axis of the parabola. (a) Locate the focus F and determine the position of the directrix and the axis. (b) Complete the drawing of the London City Hall. Key Principles: A and B are the same distance from F as they are from the... The axis is drawn... to the directrix. The vertex lies midway between the focus and the... The eccentricity of a parabola is always... B CONIC SECTIONS 15 A C