Geometric transformation problems, geometric problems for teaching

The triangle EBC and DCB have a common bottom, and EB=CD, equilateral equilateral angles, so the angle DBC=angle ECB, and because they are both angle bisectors, you can deduce the angle ABC=angle ACB, so the triangle is an isosceles triangle. Launch of AB=AC

1. Reflection transformation

The reflection transformation is the transformation of the plane to itself, if there is a straight line, so that for each point on the plane and its corresponding point, its line is bisected perpendicular by the fixed line, then this transformation is called the reflection transformation, and the fixed line is called the axis of symmetry.

The reflection transformation dissipation has the following properties:

1) Make a graph congruent with it;

2) About the symmetrical two-point line is bisected perpendicularly.

In the process of proving the problem, the reflection transformation can retain the properties of the original figure, and the Qi state makes the original dispersion conditions relatively concentrated, which is conducive to the solution of the problem.

2. Translational transformation

A translational transform is a transformation of the plane to itself, transforming any point on the plane to , such that:

1) Rays have a given direction;

2) Line segments have a given length.

This transformation is called a translational transformation.

Under a translational transformation, the graph becomes congruent and the line becomes parallel to it.

When solving geometric problems, translational transformations are often used to bring together scattered conditions with more compact positional relationships or to transform them into simpler basic figures.

3. Rotation transformation

The rotation transformation is the transformation of the plane to itself, so that the origin is transformed to itself, and any other point is transformed to , such that:

2) (Fixed Angle).

Such a transformation is called a rotational transformation, which is called the center of rotation.

The rotation transform maintains the congruence of the shape, but the orientation of the shape may vary.

In geometric solving, the function of rotation is to maintain the properties of the original figure, but change its position so that it can be combined into a new figure with a favorable argument.

The first question is as follows.

The second question is as follows.

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I wish you all the best in your work and studies

Geometric transformation is a processing method that moves image pixels relative to their spatial positions without changing the content of the image. This includes panning, mirroring, transposing, scaling, rotating, and more.

The geometric transformation of an image is to establish the mapping relationship between the source image and the transformed image.

It can be divided into forward mapping and backward mapping.

However, there are the following problems with forward mapping.

The output image coordinates in turn calculate the position of the pixel in the source image. In practice, the basic application is backward mapping.

A translational transformation is the addition of a specified vertical offset to all coordinates.

The mirror transformation is a symmetrical transformation with the midline as the axis.

width is the width of the image. It is expressed in matrix form as follows.

In MATLAB, you can use the maketform() and imtransform() functions to transform them.

Transposing an image is the exchange of position between abscissa and ordinate. The width and height of the image are reversed after the hail is held and transposed.

The mathematical expression for scaling an image is:

where sx and sy are the scaling factors. Here, a backward mapping is used.

In the process of innuendo, a floating-point coordinate pixel is generated, and it can obtain the near-source qingan value of the floating-point coordinate through a series of algorithms.

Common interpolation methods include nearest neighbor interpolation, bilinear interpolation and quadratic cubic interpolation.

During the image rotation process, the coordinates of the image need to be converted. Convert to a mathematical coordinate system with the center point as the origin. 3 steps required:

The matrix is expressed as:

where w and h are the width and height of the original image, and wnew and hnew are the width and height of the rotated image.

There is also a problem with floating-point coordinates during rotation, and it is also necessary to use nearest neighbor interpolation and bilinear interpolation.

In MATLAB, the imrotate() function is used for spinning

b = imrotate(a, angle, method, bbox)

method is the interpolation method, and bbox is used to determine whether to redefine the size.

Image affine transformation.

tform = maketform('affine', t);

g = imtransform(f, tform, interp);

Among them, interp can be:'nearest'，'bilinear'or'bicubic'。

The above content is from "Digital Image Processing: Principles and Practice (MATLAB Edition)" by Zuo Fei, Electronic Industry Press. These are study notes.

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Geometric transformations. In the solution of geometric problems, when the conditions given by the problem are not enough or not obvious, we can transform the figure into a certain answer line, which will be conducive to discovering the hidden conditions of the problem, grasping the key and essence of the problem, so that the problem can be broken through and finding a satisfactory solution Graph transformation is an important method of thinking in the book, it is a kind of idea to deal with isolated and discrete problems with a changing and moving point of view, and to understand the ideological essence of this problem solving well, and can use it accurately and reasonably. You will receive miraculous results in problem solving, and it will also effectively improve the quality of thinking

Junior high school graph transformation includes translation, folding and rotation, we need to grasp the essence of motion through experiment, operation, observation and imagination, find invariants in the motion of graphics, and then solve problems.

In the study of mathematical problems, the geometric transformation method is often used to transform the complexity problem into a simple problem and solve it.

A transformation is a one-to-one mapping of any element of a set to elements of the same set. The transformations involved in secondary school mathematics are mainly elementary transformations. For some exercises that seem difficult or even impossible to start, you can use the geometric transformation method to simplify the complex and make the difficult easy.

On the other hand, the changing perspective can also be infiltrated into the teaching of mathematics in secondary schools. Combining the study of the figure from the condition of equal rest with the study in motion is conducive to the understanding of the nature of the figure.

Geometric transformations include:

1) Translation; 2) Rotation;

3) Symmetry.

The method of solving geometric problems using the transformation of geometric figures is called the geometric transformation method.

It is the process of using the theorem in geometry to transform and push it to!

Abstract: The fold transformation is the transformation of the plane to itself, if there is a straight line l, so that for each point p on the plane and its corresponding point p, its line pp is bisected perpendicularly by the fixed line l, then this transformation is called the fold transformation, and the fixed line l is called the axis of symmetry The fold transformation has the following properties:

1) Make a graph congruent with it;

2) The two-point line with respect to l symmetry is bisected perpendicular by l

In the process of proofing, the fold transformation can retain the properties of the original figure, and make the original dispersion conditions relatively concentrated, which is conducive to the solution of the problem

You can set the length of the bottom square to be a, and the height of the cuboid is h, then the system of equations is listed.

A side + A side + H square = 9 * 9 = 81

2a square + 4ab = 144

Solve this system of equations.

It is concluded that a = 4, -6, 6, -4

Apparently the sedan repentance, a = 4, h = 7 or a = 6 h = 3 so two.