Tag: TSC Math

Reflection

Posted on by MEAN

ज्यामितिमा स्थानान्तरण लामो समयदेखि व्यापक चासोको विषय बनेको थियो। युक्लिडको आधिकारिक वा स्वयंसिद्ध सिद्धान्त नभएपनि सामान्यतया स्थानान्तरणलाई सुपरपोजिसनको सिद्धान्त (उठाउर अर्कोमा राख्नु) मा आधारित गरिएको थियो । जब १९ औं शताब्दीमा (1872 मा) फ्लेक्स क्लेनले – एर्लान्जेन विश्वविद्यालयमा – ज्यामितिमा एक नयाँ परिप्रेक्ष्य प्रस्तावित गरे तबदेखी स्थानान्तरणलाई एर्लान्जेन कार्यक्रमको रूपमा चिनिन थालिएको छ।

Transformation Geometry

Transformation Geometry is a branch of mathematics that utilize analytic geometry and algebraic function to study geometric invariants. It was introduced (developed) by Flex Klein in19th century through Erlangen programme. According to him, theer are five Kinds of Transformations:

  1. Rigid motions: Plane geometry of congruent figures, preserve distances (e.g., Euclidean transformations)
  2. Similarity transformations: The familiar geometry with similar figures.
    Similarities preserve angles and ratios between distances (e.g., resizing)
  3. Affine (matrix) transformations: Geometry of computer animation. Rectangles and parallelograms the same
    Affine transformations preserve parallelism (e.g., scaling, shear)
  4. Projective Geeometry: Geometry of photo
    Projective transformations preserve collinearity
  5. Continuous (topological) transformations: Any loop is a circle. Any path is a segment

In transformation geometry, the idea of flips, slides, and turn are key features, which will be analyzed based on its mapping. This mapping are basically of two kinds.

  1. Isometric mapping (transformation)
  2. Non-isometric mapping (transformation)

Geometric transformation involves both analytic and algebraic geometry because it can be approached using the graphical perceptive, and algebraic computational theory.

Reflection

Reflection is very common topic in physics and mathematics. When an object is place before the mirror, the image is formed behind the mirror. This is called reflection.

Definition: Reflection

Let l be a line and A be a point.
Then, reflection about the line l is denoted by T_1, which is a transformation on a plane, where

  1. if A ∉l , then Tl (A) = A’
  2. if A ∈l , then Tl(A)=A
such that
  1. the mispoint of AA’ lies on l
  2. the segment AA’is perpendiculat to l

We start a short discussion for reflection in mathematics as below.

Let us take a geometrical figures like triangle. Now, we hold the triangle vertices, one-by-one, about a line source l in a grid. Then look at the image (shadow) of the triangle on the grid. The shadow have the same size as the original triangle has, but in opposite direction.

In the above figure, ∆XYZ is a triangle and its shadow is ∆X’Y’Z’.
By actual measurements it can be seen that
∡X = ∡X’, ∡Y = ∡Y’, ∡Z = ∡Z
XY=X’Y’, YZ=Y’Z’, XZ=X’Z’
Thus, the two triangles are congruent
The important note is that

  1. l ⊥ bisector of XX’
  2. l ⊥ bisector of YY’
  3. l ⊥ bisector of ZZ’

Here l is said to be line of reflection

In summary,
  1. The image is found opposite to the mirror line l .
  2. Mirror line l is the perpendicular bisector of the line joining the object and its image.
  3. X’ is called the reflection (image) of X and so on
  4. l is called the axis of reflection.

Thus, reflection can be seen in water, in a mirror, in glass. In mathematics, an object and its reflection have the same shape and size, but the figures face in opposite directions . Also, a reflection needs

  1. Object
  2. Line of Reflection