LESSON 7 Loci Parabola JENNIFER BYRNE 2020 1

LESSON 7 Loci & Parabola JENNIFER BYRNE 2020 1

LOCI What is loci? Loci is the plural of a Locus is the route or path of an object or point as it moves under a predefined condition Drawing a circle with a compass the circumference of the circle (the lead) is the locus of the point as it rotates about the point JENNIFER BYRNE 2020 2

LOCI Problem: I have two trees planted 10 m apart and I want to plant another one but I have to take into account the span of each of my existing trees and also the span of the tree that I want to plant. Oak tree has a 8 M span and the Maple has a 6 M span. The Cherry has a 6 M span. There are only two possible positions that I can plant the Cherry Tree in can you find them? Draw Oak tree at Ø 80 mm and Maple tree at Ø 60 mm Draw their centres at 100 mm apart. Show the two possible positions for the Cherry tree Ø 60 mm JENNIFER BYRNE 2020 3

LOCI Solution: First draw in what you know then determine the loci surrounding each of the existing trees. Add 3 M to each of the existing tree spans and strike an arc The Cherry tree can be planted were the arcs intersect at these two points JENNIFER BYRNE 2020 4

DRAWING A PARABOLA BY THE LOCUS METHOD The fixed line is called the directix The point F is the focal point and lies on the axis. The Vertex V lies mid point between these two. JENNIFER BYRNE 2020 5

DRAWING A PARABOLA BY THE LOCUS METHOD Question: Trace the path that will always be equidistant from p and line ab Draw a vertical line ab. At 90° draw another line and mark point p 100 mm away. m Find point m which is midway between the line and the point JENNIFER BYRNE 2020 6

DRAWING A PARABOLA BY THE LOCUS METHOD Question: Trace the path that will always be equidistant from p and line ab Draw a line 85 mm from ab. Draw an arc 85 mm from point p Find point on curve is where they intersect. JENNIFER BYRNE 2020 7

DRAWING A PARABOLA BY THE LOCUS METHOD Question: Trace the path that will always be equidistant from p and line ab Draw a line 70 mm from ab. Draw an arc 70 mm from point p. Mark in two new points. JENNIFER BYRNE 2020 8

DRAWING A PARABOLA BY THE LOCUS METHOD Question: Trace the path that will always be equidistant from p and line ab Draw another line 60 mm from ab and so on until you have enough points to draw in the curve. JENNIFER BYRNE 2020 9

THE PARABOLA IN ARCHITECTURE The type of Loci we have just draw is also called a Parabola. The Ellipse and the Parabola are combined in many designs both in logos and in Architecture. The Sydney Harbour Bridge with the main structure being part of an Ellipse and the top section being Parabola. JENNIFER BYRNE 2020 10

THE PARABOLA IN LOGOS The Ellipse and the Parabola make up the Credit Union Logo What shapes are in the other two logos? JENNIFER BYRNE 2020 11

PARABOLA THE RECTANGLE METHOD Drawing a Parabola using the rectangle method. First draw a rectangle representing the overall size of the Parabola. Divide in half. JENNIFER BYRNE 2020 12

PARABOLA THE RECTANGLE METHOD Divide the height into 4 equal spaces. Bring a line from 1 to the Vertex. Bring a line from 2 & 3 to the Vertex. Repeat for the other side. V JENNIFER BYRNE 2020 13

PARABOLA THE RECTANGLE METHOD Divide each side of the base into 4 equal spaces. Bring each line up vertical until it hits the existing lines. Where these lines intersect are points on the Parabola. Repeat for the other side. JENNIFER BYRNE 2020 14

PARABOLA THE RECTANGLE METHOD Repeat until all lines have been drawn. Draw in the points on the Parabola. Neatly draw in the Parabola. JENNIFER BYRNE 2020 15