Consider a triangle with co-ordinates points: A(0, 0), B(3, 3) and C(2, 5). When scaling parameter is 2 towards x-axis and 4 towards y-axis, then what are the new co-ordinates of triangle?
A(0, 0), B(6, 12), C(4, 20)
Understanding Triangle Scaling and New Coordinates
The problem asks us to find the new coordinates of a triangle after applying a scaling transformation. We are given the original coordinates of the vertices of the triangle: A(0, 0), B(3, 3), and C(2, 5). We are also given the scaling parameters: 2 towards the x-axis and 4 towards the y-axis. This means the scaling factor for the x-direction ($s_x$) is 2, and the scaling factor for the y-direction ($s_y$) is 4.
Geometric Scaling Transformation
Geometric scaling is a transformation that changes the size of an object. When scaling is performed relative to the origin (0,0), the new coordinates (x', y') of a point (x, y) are calculated using the following formula:
\((x', y') = (x \cdot s_x, y \cdot s_y)\)
Where:
- \(x\) and \(y\) are the original coordinates of the point.
- \(s_x\) is the scaling factor in the x-direction.
- \(s_y\) is the scaling factor in the y-direction.
- \(x'\) and \(y'\) are the new coordinates after scaling.
In this specific problem, the scaling factors are \(s_x = 2\) and \(s_y = 4\).
Calculating New Coordinates for Each Vertex
We apply the scaling formula to each vertex of the triangle:
Scaling Vertex A(0, 0)
For point A(0, 0):
- Original coordinates: \(x = 0, y = 0\)
- Scaling factors: \(s_x = 2, s_y = 4\)
- New coordinates: \(A'(0 \cdot 2, 0 \cdot 4) = A'(0, 0)\)
The new coordinates for vertex A are (0, 0).
Scaling Vertex B(3, 3)
For point B(3, 3):
- Original coordinates: \(x = 3, y = 3\)
- Scaling factors: \(s_x = 2, s_y = 4\)
- New coordinates: \(B'(3 \cdot 2, 3 \cdot 4) = B'(6, 12)\)
The new coordinates for vertex B are (6, 12).
Scaling Vertex C(2, 5)
For point C(2, 5):
- Original coordinates: \(x = 2, y = 5\)
- Scaling factors: \(s_x = 2, s_y = 4\)
- New coordinates: \(C'(2 \cdot 2, 5 \cdot 4) = C'(4, 20)\)
The new coordinates for vertex C are (4, 20).
New Coordinates of the Scaled Triangle
After applying the scaling transformation with \(s_x = 2\) and \(s_y = 4\) to each vertex, the new coordinates of the triangle are:
- A': (0, 0)
- B': (6, 12)
- C': (4, 20)
Thus, the new coordinates of the triangle are A(0, 0), B(6, 12), and C(4, 20).
| Original Vertex | Original Coordinates (x, y) | Scaling Factors (\(s_x\), \(s_y\)) | New Coordinates (\(x \cdot s_x\), \(y \cdot s_y\)) | New Vertex |
|---|---|---|---|---|
| A | (0, 0) | (2, 4) | (\(0 \cdot 2\), \(0 \cdot 4\)) = (0, 0) | A' |
| B | (3, 3) | (2, 4) | (\(3 \cdot 2\), \(3 \cdot 4\)) = (6, 12) | B' |
| C | (2, 5) | (2, 4) | (\(2 \cdot 2\), \(5 \cdot 4\)) = (4, 20) | C' |
Revision Table: Triangle Scaling Basics
| Concept | Description | Formula (relative to origin) |
|---|---|---|
| Scaling | Changing the size of an object. | \((x', y') = (x \cdot s_x, y \cdot s_y)\) |
| Uniform Scaling | \(s_x = s_y\) (Size changes, shape preserved) | \((x', y') = (x \cdot s, y \cdot s)\) |
| Non-Uniform Scaling | \(s_x \neq s_y\) (Size and shape may change) | \((x', y') = (x \cdot s_x, y \cdot s_y)\) |
| Scaling Factors | Values \(s_x, s_y\) determining the extent of scaling. | \(s_x > 1\) for enlargement in x, \(0 < s_x < 1\) for shrinking in x. Same for \(s_y\). |
Additional Information: Types of Transformations
Geometric transformations are operations that change the position, size, or shape of an object. Common 2D transformations include:
- Translation: Moving an object to a new location without changing its size or orientation. \( (x', y') = (x + t_x, y + t_y) \)
- Rotation: Rotating an object about a point (usually the origin) by a certain angle.
- Scaling: Changing the size of an object. Can be uniform or non-uniform.
- Shearing: Skewing an object along an axis.
Scaling is a fundamental transformation in computer graphics and geometry.
Important Questions from Two Dimensional Geometric Transformation
- The construction of computer-generated simulations that simulate real-world activities using interactive graphics software and hardware is called:
- In order to move objects in computer graphics that generate animation effects which of the following is necessary?
Consider a triangle represented by A(0, 0), B(1, 1), C(5, 2). The triangle is rotated by 45 degrees about a point P(–1, –1). The co-ordinates of the new triangle obtained after rotation shall be _______
Consider a unit square centred at origin. The coordinates of the square are translated by a factor \(\left(\frac{1}{2}, 1\right)\) and rotated by an angle of 90°. What shall be the coordinates of the new square ?
Consider a triangle A(0,0), B(1,1) and C(5,2). The triangle has to be rotated by an angle of 45° about the point P(-1, -1). What shall be the coordinates of the new triangle ?
