Surface line
As surface lines is referred to in the descriptive geometry the generating lines of different surfaces :
- The surface lines of a conical surface pass through a fixed point (tip) and pass through the points of a curve (guide curve). Compare the illustration at the end of the introduction.
- In the case of a cylinder surface, the surface lines that run through the guide curve are parallel .
- The generators of the tangent surface of a space curve , i.e. H. the tangents to the curve are called surface lines.
In the case of lateral surfaces of bodies , especially in school geometry, the distances between the base and top surface or point are called surface lines. In the special cases considered at school, these lie on the surface lines straight in the sense of descriptive geometry.
Descriptive geometry
In descriptive geometry, the straight lines that create an (initially unrestricted) lateral surface, as described in the introduction, are referred to as lateral lines. A restricted section of this straight line is then actually drawn. The surface lines are important here
- for the construction of the contour of a cone-like body and thus also for the determination of visible or hidden parts of an opaque body (the figure at the end of the introduction indicates hidden parts of the contour with dashed lines),
- for the construction of the shadow that such a body casts with defined lighting.
Elementary and school geometry
Here, as a rule, only lines on three specific types of bodies of revolution are referred to as surface lines. These bodies of revolution are the straight circular cone , the rotationally symmetrical truncated cone and the straight circular cylinder . In these cases, all surface lines are congruent and therefore of the same length. Often their common length is simply referred to as the “surface line”, but this article always distinguishes between the lines and their length.
Straight circular cone
Every connection between a point on the base circle and the tip is a surface line. With their sections, these are the only sections that lie entirely on the lateral surface. Other connecting lines from two points of the jacket are always inside the cone body. The length s of a surface line is the radius of the sector of a circle that the developed surface forms, compare the figure on the right.
Rotationally symmetrical truncated cone
A rotationally symmetrical truncated cone is the body that emerges from a straight circular cone when a cone is cut off between the base and the tip with a cutting plane parallel to the base. The straight line connecting the centers of the base and top surface is the axis of symmetry of the truncated cone. Each plane containing this axis intersects the jacket in a surface line and all surface lines lie on such a plane through the axis. The surface lines with their sub-sections are the only sections that lie entirely on the surface of the jacket. Other connecting lines from two points of the mantle are always inside the truncated cone.
Straight circular cylinder
As with the truncated cone, the straight line connecting the center points of the base and top surface is the axis of symmetry of the cylinder. Each plane containing this axis intersects the jacket in a surface line and all surface lines lie on such a plane through the axis. The surface lines with their sub-sections are the only sections that lie entirely on the surface of the jacket. Other connecting lines from two points of the jacket are always inside the circular cylinder.
The length h of a surface line corresponds to the length of the height of the cylinder. Two sides of the developed jacket, a rectangle , are jacket lines, the other two sides are the base or cover circle line. Compare the picture on the right.
Web links
The English term slant height comes close to the term "surface line of a straight circular cone" in the sense of a line length as it is used in school mathematics for straight circular cones.
- Slant height of a right cone : English explanation of the English term in Math Open Reference , with an interactive Java applet .
literature
- Rolf Baumann: Geometry for the 9./10. Class . Centric stretching, Pythagorean theorem, circle and body calculations. 4th edition. Mentor-Verlag, Munich 2003, ISBN 3-580-63635-9 , pp. 95 ff . (Surface line in the school geometry).
- Reinhold Müller: Guide for lectures on descriptive geometry . 2nd Edition. Vieweg, Braunschweig 1903, p. 32 f . ( [1.Ex.pdf PDF] [accessed on May 24, 2013]).
- Karl Rohn and Erwin Papperitz: Textbook of Descriptive Geometry . 1st edition. Salzwasser GmbH, Paderborn 1911, p. 362 ff . ( Book on Google Books with access to extracts [accessed May 24, 2013]).