Eccentricity is called numerical characteristic of conic section (the figure which is turning out when crossing the plane and cone). Eccentricity does not change at the movement of the plane and also transformations of similarity (change of the sizes at preservation of a form). Figuratively speaking eccentricity is characteristic of a form ("flattening", in case of an ellipse) figures, but not its size.
It is required to you
- - compasses;
- - ruler;
- - calculator.
Instruction
1. If are set focus and the head mistress of conic section, then for finding of eccentricity use definition of this class of figures. All nondegenerate conic sections (except for a circle) can be constructed in the following way: - choose a point and a straight line on the plane; - set real positive number e; - note all points for which the distance to the chosen point and to a straight line differs in e time.
2. At the same time the chosen point will be called focus of conic section, a straight line - the head mistress, and number e - eccentricity. Depending on number size e, four types of conic sections turn out: - at e1 – a hyperbole; - at e =0 – a circle (conditionally).
3. Proceeding from definition to find the eccentricity of conic section: - choose any point on this figure; - measure distance from this point to section focus; - measure distance from this point to the head mistress (for this purpose, lower a perpendicular on the head mistress and define a point of intersection of the head mistress and a perpendicular); - divide distance from a point to focus into distance from a point to the head mistress.
4. If lengths of big and small axes of an ellipse (its "length" and "width") are known, then for calculation of eccentricity use the following formula: е = √ (1²/A²), where and, And – lengths of small and big axes (or half shafts), respectively.
5. If under the terms of a task radiuses of an apocentre and pericenter of an ellipse are set, then to find eccentricity, apply the following formula: е = (Ra-Rp)/(Ra+Rp), where Ra and Rp – radiuses of an apocentre and pericenter of an ellipse, respectively (radius of an apocentre is called the distance from ellipse focus to the most remote point; radius of a pericenter is called the distance from ellipse focus to the least remote point).
6. If are known distance between focuses of an ellipse and length of its bigger axis, then for calculation of eccentricity just divide distance between focuses into axis length: е = f/A, where f – distance between ellipse focuses.