Example Helical Surface of Helical Tangent Line Additional
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Example: Helical Surface of Helical Tangent Line 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: Helical Surface of Helical Tangent Line A helix is given by its start point P 0 and end point P 5. Construct a helical surface, which is generated by applying a helical motion to a tangent of this helix. The helical surfaces is cut by horizontal level plane π1. Define the remaining helical surface. P 5’’ P 4’’ Complete the helix by points and tangent lines. P 3’’ Divide the given height and the turn into (five) equal parts. P 2’’ P 1’’ P 0’’ π1‘‘ P 5’ P 4’ P 3’ P 0’ P 1’ P 2’ 2
Example: Helical Surface of Helical Tangent Line A helix is given by its start point P 0 and end point P 5. Construct a helical surface, which is generated by applying a helical motion to a tangent of this helix. The helical surfaces is cut by horizontal level plane π1. Define the remaining helical surface. P 5’’ P 4’’ Complete the helix by points and tangent lines. P 3’’ P 2’’ h/2 P 1’’ Considering that 22/7 exceeds π the following ratios are valid: P 0’’ π1‘‘ P 5’ P 4’ P 3’ 3, 5 P 0’ P 1’ P 2’ p h/2 11 3
Example: Helical Surface of Helical Tangent Line A helix is given by its start point P 0 and end point P 5. Construct a helical surface, which is generated by applying a helical motion to a tangent of this helix. The helical surfaces is cut by horizontal level plane π1. Define the remaining helical surface. P 5’’ P 4’’ Complete the helix by points and tangent lines. P 3’’ P 2’’ h/2 P 1’’ Considering that 22/7 exceeds π the following ratios are valid: P 0’’ π1‘‘ P 5’ P 4’ Use the cone to construct tangent lines through given points as parallel lines to according generatrices of cone. P 3’ 3, 5 P 0’ P 1’ P 2’ p h/2 11 4
Example: Helical Surface of Helical Tangent Line A helix is given by its start point P 0 and end point P 5. Construct a helical surface, which is generated by applying a helical motion to a tangent of this helix. The helical surfaces is cut by horizontal level plane π1. Define the remaining helical surface. Determine the helical surface between the helix and the basement. Define intercepts of tangent lines with the basement (as far as possible). h/2 The intersection curve is an EVOLVENT– that means, • It can be obtained by tracing the free end of a taunt string, when the other end is wound on the circle. • tangent line of an evolvent is perpendicular to generatrix. P 5’’ P 4’’ P 3’’ P 2’’ P 1’’ P 0’’ π1‘‘ P 5’ P 4’ P 3’ 3, 5 P 0’ u8 P 1’ P 2’ u2 3*(u8) p h/2 11 5
Example: Helical Surface of a Tangent Line of a Helix A helix is given by its start point P 0 and end point P 5. Construct a helical surface, which is generated by applying a helical motion to a tangent of this helix. The helical surfaces is cut by horizontal level plane π1. Define the remaining helical surface. Determine the helical surface between the helix and the basement. Define intercepts of tangent lines with the basement (as far as possible). h/2 The intersection curve is an EVOLVENT– use its properties: • It can be obtained by tracing the free end of a taunt string, when the other end is wound on the circle. • tangent line of an evolvent is perpendicular to generatrix. • the circle is its evolute = locus of midpoints of curvature 3, 5 P 5’’ P 4’’ P 3’’ P 2’’ P 1’’ P 0’’ π1‘‘ P 5’ P 4’ P 3’ P 0’ u8 P 1’ P 2’ u2 3*(u8) p h/2 11 6
Example: Helical Surface of Helical Tangent Line A helix is given by its start point P 0 and end point P 5. Construct a helical surface, which is generated by applying a helical motion to a tangent of this helix. The helical surfaces is cut by horizontal level plane π1. Define the remaining helical surface. P 5’’ P 4’’ Determine the helical surface between the helix and the basement. Show proper visibility. P 3’’ P 2’’ h/2 P 1’’ P 0’’ π1‘‘ P 5’ P 4’ P 3’ P 0’ u8 P 1’ P 2’ u2 3*(u8) 7
