Rotation Formulas
Shared 11/27/2025•87 views
Rotation (परिक्रमण) S.N. Angle of rotation परिक्रमणको कोण Centre of rotation परिक्रमणको केन्द्र Object वस्तु Image प्रतिबिम्ब (i) $+90^\circ$ or $-270^\circ$ $(0, 0)$ $P(x, y)$ $P'(-y, x)$ (ii) $-90^\circ$ or $+270^\circ$ $(0, 0)$ $P(x, y)$ $P'(y, -x)$ (iii) $\pm 180^\circ$ $(0, 0)$ $P(x, y)$ $P'(-x, -y)$ (iv) $+90^\circ$ or $-270^\circ$ $(a, b)$ $P(x, y)$ $P'(-y+a+b, x-a+b)$ (v) $-90^\circ$ or $+270^\circ$ $(a, b)$ $P(x, y)$ $P'(y+a-b, -x+a+b)$ (vi) $\pm 180^\circ$ $(a, b)$ $P(x, y)$ $P'(-x+2a, -y+2b)$ Notes on Rotation (परिक्रमण सम्बन्धी टिप्पणीहरू) Rotation is a rigid transformation, meaning it preserves size, shape, and orientation of the object. The center of rotation is the fixed point around which the object rotates. The angle of rotation specifies the amount and direction of rotation. A positive angle usually indicates counter-clockwise rotation, while a negative angle indicates clockwise rotation. When the center of rotation is the origin $(0,0)$, the formulas are simplified. When rotating around a point $(a,b)$ other than the origin, it can be thought of as a three-step process: Translate the object so that the center of rotation $(a,b)$ moves to the origin. If $P(x,y)$ is translated, it becomes $P(x-a, y-b)$. Perform the rotation about the origin using the respective formulas. Translate the object back by moving the origin back to $(a,b)$. A rotation of $360^\circ$ (or its multiples) maps an object onto itself. A rotation of $180^\circ$ about the origin is equivalent to point reflection through the origin. Combined Rotation (संयुक्त परिक्रमण) If $R_1$ and $R_2$ are two rotations about the same center, then combined rotation: $(R_1 \circ R_2) = (R_2 \circ R_1) = R_1 + R_2$ If $R_1 = [(0, 0), \theta_1]$ and $R_2 = [(0, 0), \theta_2]$ are two rotations, then the combined rotation: $R_2 \circ R_1 = R_1 \circ R_2 = R[(0, 0), (\theta_1 + \theta_2)]$