A concave mirror is often used behind the bulb in a lamp, searchlight or projector to give a parallel beam of light but if the aperture of the mirror is too big the beam will spread out. For this reason parabolic mirrors are used, these have a slightly different shape to the spherical mirrors and will give a perfectly parallel beam. They are also used in all big reflecting telescopes because they give a much sharper image than a spherical mirror. (In the following diagrams the difference in the shape of the parabolic and spherical mirrors would be too small to see so both the spreading and the mirror shapes have been exaggerated!)

AbstractThe paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.

A concave mirror will converge a beam of light and it gives a real image. However, if the object is closer to the mirror than its focal length the image is virtual. The focal length and radius of curvature of a concave mirror are real.The image produced is up the right way, virtual and magnified if the object is closer to the mirror than its focal length but inverted and real if it is further away. Uses of concave mirrorsShaving mirrors, make-up mirrors, dentists' mirrors, microscopes, fun mirrors, lamp reflectors, reflecting telescope.

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Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question.The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

When you look into the two sides of the bowl of a spoon you are actually using two simple curved mirrors. The side that curves inwards is called CONCAVE and the side that curves outwards is called CONVEX.It is the same with mirrors, if the reflecting surface curves inwards you have a CONCAVE mirror and if the reflecting surface curves outwards you have a CONVEX mirror.Many curved mirrors are parts of a sphere and so are known as SPHERICAL MIRRORS. The centre of this sphere is called the CENTRE OF CURVATURE (C) of the mirror and its radius the RADIUS OF CURVATURE (R) of the mirror.The distance from the pole (P) to the principal focus (F) is called the FOCAL LENGTH of the mirror. The principal focus and focal length of a concave mirror are real but those of a convex mirror are virtual.The effect of the two types of curved mirror on a parallel beam of light is shown by the two diagrams.

A convex mirror will diverge a beam of light and it gives a virtual image.The focal length and radius of curvature of a convex mirror are virtual. The image produced is always up the right way and smaller than the object, the convex mirror gives a wide field of view because of this. Uses of convex mirrorsWide angle car wing mirrors, buses' mirrors, security mirrors in shops, fun mirrors.

A method for designing an optical element with two free-form surfaces generating a prescribed illuminance distribution in the case of an extended light source is considered. The method is based on the representation of the optical element surfaces by bicubic splines and on the subsequent optimization of their parameters using a quasi-Newton method implemented in the Matlab software. To calculate the merit function, a version of the ray tracing method is proposed. Using the proposed method, an optical element with record characteristics was designed: the ratio of the element height to the source size is 1.6; luminous efficiency is 89.1 %; uniformity of the generated distribution (the ratio of the minimum and average illuminance) in a given square region is 0.92.