Example Cylinder Cut by Cylinder Locomotive Additional Task

Example: Cylinder Cut by Cylinder - Locomotive 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: Cylinder Cut by Cylinder - Locomotive b) The given image of a toy (locomotive) shows the union of two cylinders. Determine the intersection curve. The profile of the chimney (red coloured) is situated within the horizontal cylinder (blue coloured) and defines a smaller cylinder. The intersection curve of these two cylinders consists of 2 congruent spatial curves. In this example the upper spatial curve is the only required one. z Use a vertical auxiliary plane, which is parallel to both rotational axes of the cylinder. Such an auxiliary plane intersects each cylinder along generatrices. Choose an auxiliary plane through the almost left point (generatrix) of the chimney. This plane is also tangent plane of the vertical cylinder and therefore the intercept defines an outline point of the chimney. The according (outline) generatrix defines the tangent line. x y 2

Example: Cylinder Cut by Cylinder - Locomotive b) The given image of a toy (locomotive) shows the union of two cylinders. Determine the intersection curve. Choose further auxiliary plane through intercept of rotational axes. z The intercepts of the intersection curve are extrema, therefore their tangent lines are horizontal. Another explanation: The tangent planes through these intercepts are parallel the y- axis and intersect along parallel line with respect to the yaxis. y x 3

Example: Cylinder Cut by Cylinder - Locomotive b) The given image of a toy (locomotive) shows the union of two cylinders. Determine the intersection curve. Choose further auxiliary plane through the given point with its tangent line. z Construction of tangent lines: a tangent line of the intersection curve is the intersection line of according tangential planes (of cylinder). Define the intersection line of both tangential planes with a plane (e. g. parallel to yz-plane). Determine the intersection line of the tangential plane of the vertical cylinder with chosen plane. The intersection line of the tangential plane of the horizontal cylinder with the chosen plane is the tangent line of the level circle. The required tangent line must go through the intercept of these intersection lines. Use symmetry to define the tangent line through the second point. y x 4

Example: Cylinder Cut by Cylinder - Locomotive b) The given image of a toy (locomotive) shows the union of two cylinders. Determine the intersection curve. Choose further auxiliary plane through the almost right point (generatrix) of the chimney. z This auxiliary plane is tangent plane of the chimney. Hence the outline generatrix defines the tangent line. y x 5

Example: Cylinder Cut by Cylinder - Locomotive b) The given image of a toy (locomotive) shows the union of two cylinders. Determine the intersection curve. For accuracy choose further auxiliary plane. z Connect intercepts of the intersection curve and show proper visibility. y x 6

Example: Cylinder Cut by Cylinder - Locomotive b) The given image of a toy (locomotive) shows the union of two cylinders. Determine the intersection curve. For accuracy choose further auxiliary plane. z Connect intercepts of the intersection curve and show proper visibility. y x 7