Merge pull request #28422 from Shrutikasri04:first-pr

docs(calib3d): fix grammar and clarify radial distortion formula in camera calibration tutorial #28422

This PR improves the camera calibration tutorial documentation by:

- Fixing minor grammatical issues for better readability.
- Clarifying the radial distortion model by explicitly defining r² = x² + y².

These changes do not affect functionality and are intended to make the mathematical explanation clearer for readers and new users.

doc/tutorials/calib3d/camera_calibration/camera_calibration.markdown

@@ -12,9 +12,9 @@ Camera calibration With OpenCV {#tutorial_camera_calibration}
 | Compatibility | OpenCV >= 4.0 |
 
 
-Cameras have been around for a long-long time. However, with the introduction of the cheap *pinhole*
+Cameras have been around for a very long time. However, with the introduction of the cheap *pinhole*
 cameras in the late 20th century, they became a common occurrence in our everyday life.
-Unfortunately, this cheapness comes with its price: significant distortion. Luckily, these are
+Unfortunately, this low cost comes with a trade-off: significant distortion. Luckily, these are
 constants and with a calibration and some remapping we can correct this. Furthermore, with
 calibration you may also determine the relation between the camera's natural units (pixels) and the
 real world units (for example millimeters).
@@ -22,8 +22,10 @@ real world units (for example millimeters).
 Theory
 ------
 
-For the distortion OpenCV takes into account the radial and tangential factors. For the radial
-factor one uses the following formula:
+For distortion, OpenCV takes into account both radial and tangential factors. For the radial
+factor one uses the following formulas:
+
+\f[r^2 = x^2 + y^2\f]
 
 \f[x_{distorted} = x( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6) \\
 y_{distorted} = y( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6)\f]
@@ -64,10 +66,10 @@ objects. Currently OpenCV supports three types of objects for calibration:
 -   Symmetrical circle pattern
 -   Asymmetrical circle pattern
 
-Basically, you need to take snapshots of these patterns with your camera and let OpenCV find them.
+In practice, you need to capture multiple images of these patterns using your camera and let OpenCV find them.
 Each found pattern results in a new equation. To solve the equation you need at least a
 predetermined number of pattern snapshots to form a well-posed equation system. This number is
-higher for the chessboard pattern and less for the circle ones. For example, in theory the
+higher for the chessboard pattern and lower for circle-based patterns.For example, in theory the
 chessboard pattern requires at least two snapshots. However, in practice we have a good amount of
 noise present in our input images, so for good results you will probably need at least 10 good
 snapshots of the input pattern in different positions.
@@ -148,8 +150,8 @@ Explanation
     is defined in squares, but detection result is list of inner corners and that's why is smaller
     by 1 in both dimensions.
 
-    Then again in case of cameras we only take camera images when an input delay time is passed.
-    This is done in order to allow user moving the chessboard around and getting different images.
+    In the case of live cameras, we only capture images when an input delay time is passed.
+    This is done to allow the user to move the chessboard around and getting different images.
     Similar images result in similar equations, and similar equations at the calibration step will
     form an ill-posed problem, so the calibration will fail. For square images the positions of the
     corners are only approximate. We may improve this by calling the @ref cv::cornerSubPix function.
@@ -160,6 +162,7 @@ Explanation
     visualization feedback purposes we will draw the found points on the input image using @ref
     cv::findChessboardCorners function.
     @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp pattern_found
+
 -#  **Show state and result to the user, plus command line control of the application**
 
     This part shows text output on the image.
@@ -170,6 +173,7 @@ Explanation
     Then we show the image and wait for an input key and if this is *u* we toggle the distortion removal,
     if it is *g* we start again the detection process, and finally for the *ESC* key we quit the application:
     @snippet samples/cpp/tutorial_code/calib3d/camera_calibration/camera_calibration.cpp await_input
+
 -#  **Show the distortion removal for the images too**
 
     When you work with an image list it is not