Example A True Shape of a Planar Figure

Example A: True Shape of a Planar Figure – Auxiliary View and Rotation 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 A: True Shape of a Planar Figure – Auxiliary View and Rotation Parallel projection pictures a PLANAR figure undistorted, if the planar figure lies parallel to an image plane. a) Triangle ABC lies in a plane, which shows second edge view. Rotate plane ABC about an axis, which is parallel to p 1 and is imaged as a point in the front view. e. g: axis through point B. Thus the rotated triangle shows true shape and size in the plan view. A‘‘ or e. g. rotational axis through C A 0‘‘ C 0‘ C‘ A‘ B 0‘‘ C‘‘ =C 0‘‘ e‘‘ A 0‘ A‘‘ e‘‘ B‘‘ =B 0‘‘ B‘‘ p 2‘ = p 1‘‘ e‘ C‘ =C 0‘ A 0‘ B‘ =B 0‘ A‘ p 2‘ = p 1‘‘ e‘ B‘ B 0‘ 2

Example B: True Shape of a Planar Figure – Auxiliary View and Rotation Use an auxiliary view to image the plane ABC as an edge. For more details see file: „Project a Plane as an Edge“ C’’ 1’’ A’’ h 1’’ B’’ p 1’’=p 2’ C’ 1’ p 1 ’’’= A’ p 3 ’ h 1’ C’’’ e’’’ A’’’=1’’’=h 1’’’ B’ B’’’ 3

Example B: True Shape of a Planar Figure – Auxiliary View and Rotation Rotate the plane e = (ABC) parallel to p 1. The figure shows true size and shape in the plan view. C’’ h 1’’ A’’ B’’ p 1’’=p 2’ C’ C 0’’’ A’ 1’ p 1 ’’’= p 3 ’ h 1’ C’’’ s 1‘‘‘ A’’’=1’’’=h 1’’’ B’ B’’’ B 0’’’ 4

Example B: True Shape of a Planar Figure – Auxiliary View and Rotation Rotate the plane e = (ABC) parallel to p 1. The figure shows true size and shape in the plan view. Draw respective folding lines. The circles of rotation C → C 0 and B → B 0 lie parallel to p 3 and thus the circles are imaged as circles in the auxiliary view. In the plan view these circles are imaged as edges. Check your accuracy: The triangle ABC must intersect the rotated C 0’’’ triangle A 0 B 0 C 0 in point 1 (which lies on the principle line h 1), because the principle line h 1 is the C’’’ axis of rotation. C’’ h 1’’ A’’ B’’ p 1’’=p 2’ C 0’ C’ A’ 1’ p 1 ’’’= p 3 ’ h 1’ e’’’ A’’’=1’’’=h 1’’’ B’ B 0’ s 1‘‘‘ B’’’ B 0’’’ 5

Example B: True Shape of a Planar Figure – Auxiliary View and Rotation The construction can be simplified as follows: C’’ h 1’’ A’’ B’’ C 0’ p 1’’=p 2’ C’ C 0’’’ p 1 ’’’= p 3 ’ h 1’ A’ 1’ C’’’ e’’’ A’’’=1’’’=h 1’’’ B’ B 0’ s 1‘‘‘ B’’’ B 0’’’ 6

Example: True Shape of a Planar Figure – Auxiliary View and Rotation C’’ h 1’’ A’’ Short version: Construction of true size and shape of a planar figure about a principle line (axis of rotation). B’’ p 1’’=p 2’ C 0’ C’ h 1’ A’ 1’ B’ B 0’ 7