Example Intersection of Cylinder with Plane Development Additional

Example: Intersection of Cylinder with Plane- Development Additional Task to the multimedia book „Darstellende Geometrie/ 3 D-Geometry“, published by Veritas Educational Edition Student Edition ISBN - 978 -3 -7058 -9079 -4 ISBN - 978 -3 -7058 -9293 -4 Special edition for teachers: The print version shows handouts of theory and worked-out examples. Each handout can be printed in colour and is also suitable as a solutions handout. Through the use of animated Power. Point files it is possible to structure the lessons in a contemporary and innovative manner for students. Special edition for students: The print version consists of well prepared worksheets to start working right away. On the CD you will find colourful Power. Point presentations, including theory as well as solved examples with step-by-step explanations. This provides a highly efficient technique in developing an understanding of geometry and its concepts. for more DETAILS and ORDER 1

Example: Intersection of Cylinder with Plane- Development A truncated cylinder is given by its plan and front view. Complete the profile view of the truncated cylinder. Determine the true size and shape of the top face (= intersection curve) and define the development of the truncated cylinder. ellipse e The intersection curve is …………. The midpoint M 2 of the ellipse lies on the rotational axis of the cylinder. A‘‘‘ D‘‘‘ M 2‘‘‘ e‘‘ C‘‘‘ M 2‘‘= C‘‘= D‘‘ Locate M 2 in the profile view. A and B are the major vertices of the ellipse. B‘‘‘ M 1‘‘ C‘ Define minor vertices C and D. Construct the intersection curve (= ellipse) in the profile view. Define the outline generatrices in the profile view. A‘ M 1‘ = M 2‘ p 1‘‘ = p 2‘ B‘ e‘ D‘ 2

Example: Intersection of Cylinder with Plane- Development A truncated cylinder is given by its plan and front view. Complete the profile view of the truncated cylinder. Determine the true size and shape of the top face (= intersection curve) and define the development of the truncated cylinder. AIV A‘‘‘ Determine the true size and shape of the intersection curve by using a fourth auxiliary view. Transfer appropriate distances (of the plan view to the fourth view). D‘‘‘ M 2‘‘‘ DIV M 2 IV e‘‘ C‘‘‘ M 2‘‘= C‘‘= D‘‘ s 4‘‘ B‘‘‘ M 1‘‘ C‘ A‘ BIV CIV M 1‘ = M 2‘ p 2‘‘‘ = p 4‘‘ p 1‘‘ = p 2‘ B‘ e‘ D‘ 3

Example: Intersection of Cylinder with Plane- Development A truncated cylinder is given by its plan and front view. Complete the profile view of the truncated cylinder. Determine the true size and shape of the top face (= intersection curve) and define the development of the truncated cylinder. AIV A‘‘‘ D‘‘‘ M 2‘‘‘ DIV M 2 IV e‘‘ C‘‘‘ M 2‘‘= C‘‘= D‘‘ B‘‘‘ M 1‘‘ C‘ A‘ 3, 5 BIV CIV M 1‘ = M 2‘ p 2‘‘‘ = p 4‘‘ p 1‘‘ = p 2‘ B‘ e‘ 2 2 r 1 D‘ p 2 p 6, 28 11 U= 2 rp 4

Example: Intersection of Cylinder with Plane- Development A truncated cylinder is given by its plan and front view. Complete the profile view of the truncated cylinder. Determine the true size and shape of the top face (= intersection curve) and define the development of the truncated cylinder. AIV Divide the circumference into eight equal parts. A‘‘‘ D‘‘‘ M 2‘‘‘ DIV M 2 IV e‘‘ C‘‘‘ M 2‘‘= C‘‘= D‘‘ B‘‘‘ M 1‘‘ C‘ A‘ 2 r BIV CIV M 1‘ = M 2‘ p 2‘‘‘ = p 4‘‘ p 1‘‘ = p 2‘ B‘ e‘ 2 D‘ 2 p 6, 28 2 rp 5

Example: Intersection of a Cylinder with a Plane- Development A truncated cylinder is given by its plan and front view. Complete the profile view of the truncated cylinder. Determine the true size and shape of the top face (= intersection curve) and define the development of the truncated cylinder. AIV Define generatrices through these points (initial and final point are identical) and transfer appropriate distances of the lateral surface. A‘‘‘ D‘‘‘ M 2‘‘‘ DIV M 2 IV e‘‘ C‘‘‘ M 2‘‘= C‘‘= D‘‘ B‘‘‘ M 1‘‘ C‘ A‘ 2 r BIV CIV M 1‘ = M 2‘ p 2‘‘‘ = p 4‘‘ p 1‘‘ = p 2‘ B‘ e‘ 2 D‘ 2 p 6, 28 2 rp 6

Example: Intersection of Cylinder with Plane- Development A truncated cylinder is given by its plan and front view. Complete the profile view of the truncated cylinder. Determine the true size and shape of the top face (= intersection curve) and define the development of the truncated cylinder. AIV Connect according points of the development. A‘‘‘ Notice: The development of the intersection curve (= ellipse) of a cylinder with a plane is always a sine curve. D‘‘‘ M 2 IV e‘‘ C‘‘‘ BIV CIV M 2‘‘= C‘‘= D‘‘ B‘‘‘ M 1‘‘ C‘ A‘ 2 r DIV M 1‘ = M 2‘ p 2‘‘‘ = p 4‘‘ p 1‘‘ = p 2‘ B‘ e‘ 2 D‘ 2 p 6, 28 2 rp 7