Question: A mirror is tilted with an angle theta to the vertical, and observor of height h is standing in fron...

Question

A mirror is tilted with an angle theta to the vertical, and observor of height h is standing in front of the mirror perpendicularly. what is the minimum height of mirror needed for observor to view a full body image of his

Answer 1

Solution

To determine the minimum height of the mirror needed for an observer to view their full body image, we can use the principle of reflection and ray diagrams.

Let the observer's height be $H$ . Let the observer's eyes be at height $H_e$ from the ground. The top of the head is at height $H$ , and the feet are at height 0.

To see the top of the head, a ray of light from the top of the head must reflect off the mirror and enter the observer's eyes. Let the top of the mirror required for this be at point $M_T$ . To see the feet, a ray of light from the feet must reflect off the mirror and enter the observer's eyes. Let the bottom of the mirror required for this be at point $M_B$ .

Consider the reflection of the top of the head. Let the top of the head be at point T (height $H$ ). The eyes are at point E (height $H_e$ ). According to the law of reflection, the angle of incidence equals the angle of reflection. This means the ray behaves as if it came from the image of the top of the head, T', which is located behind the mirror. For a plane mirror, the image is as far behind the mirror as the object is in front, and the line segment connecting the object and image is perpendicular to the mirror surface.

Let's first consider the standard case where the mirror is vertical. Let the observer be at a horizontal distance $d$ from the vertical mirror. The image T' is at height $H$ and at a horizontal distance $d$ behind the mirror. The ray from T reflects at $M_T$ and goes to E. The points T', $M_T$ , and E are collinear. Consider the vertical positions: E is at height $H_e$ , T' is at height $H$ . $M_T$ is on the mirror. By similar triangles (or by considering the line segment T'E intersecting the vertical mirror plane), the height of $M_T$ is the average of the heights of T' and E, assuming the horizontal distances from the mirror are equal for T' and E. Height of $M_T = (H + H_e) / 2$ .

Similarly, consider the reflection of the feet. Let the feet be at point F (height 0). The image F' is at height 0 and at a horizontal distance $d$ behind the mirror. The ray from F reflects at $M_B$ and goes to E. The points F', $M_B$ , and E are collinear. Consider the vertical positions: E is at height $H_e$ , F' is at height 0. $M_B$ is on the mirror. By similar triangles (or by considering the line segment F'E intersecting the vertical mirror plane), the height of $M_B$ is the average of the heights of F' and E. Height of $M_B = (0 + H_e) / 2 = H_e / 2$ .

The minimum height of the mirror required to see the full body is the vertical distance between $M_T$ and $M_B$ . Minimum mirror height = Height of $M_T$ - Height of $M_B$ Minimum mirror height = $(H + H_e) / 2 - H_e / 2 = H / 2$ .

This result is independent of the observer's distance from the mirror and the height of the eyes, provided the mirror is a plane mirror and is oriented vertically.

Now, let's consider the effect of tilting the mirror by an angle $\theta$ to the vertical. For a plane mirror, the relationship between an object and its image is such that the mirror surface is the perpendicular bisector of the line segment connecting the object point and its image point. This geometric relationship holds regardless of the mirror's orientation. The ray from the object point to the mirror and then to the observer's eye follows the path that a ray from the image point to the eye would take in a straight line. Therefore, to see the full image of the observer (from height 0 to H), the mirror must span the region where rays from the full height of the image (from height 0 to H) intersect the mirror surface on their way to the observer's eyes (at height $H_e$ ).

Let the observer be at point O. Let the image of the observer be O'. For a plane mirror, the image O' is a virtual image located behind the mirror. The size of the image is the same as the object. The line segment connecting any point on the object to its corresponding point on the image is perpendicular to the mirror surface and is bisected by the mirror surface. Consider the observer standing vertically. The image will also be oriented vertically relative to the object, but its position and orientation in space depend on the mirror's position and orientation. Let's consider the observer's eye E. To see the image of the top of the head T', the ray from T' to E must hit the mirror. To see the image of the feet F', the ray from F' to E must hit the mirror. The minimum extent of the mirror needed is the portion that reflects the light rays from the extreme points (top of head and feet) into the eyes.

Crucially, the path of the light ray from the object point to the mirror and then to the eye is the same as the path of a straight line ray from the image point to the eye. The location on the mirror where the reflection occurs is simply the intersection of the line segment connecting the image point and the eye with the mirror surface.

The height of the image of the observer is still $H$ . The eyes of the observer are at a fixed position relative to the observer's body. The required section of the mirror is determined by the lines of sight from the observer's eyes to the top and bottom of their image. Let's consider the vertical extent of the image relative to the eyes. The top of the image is a height $H - H_e$ above the eye level of the image, and the bottom of the image is a height $H_e - 0 = H_e$ below the eye level of the image. The total vertical extent of the image relative to the eye level is $(H - H_e) + H_e = H$ .

The mirror must cover the range of reflection points for rays originating from the entire image and terminating at the observer's eyes. The size of this range on the mirror is determined by the geometry of similar triangles formed by the eye, the image points, and the mirror.

Regardless of the tilt of the plane mirror, the image formed by a plane mirror is always the same size as the object. The required portion of the mirror to view the entire image from a specific point (the eye) is determined by the angular size of the image as seen from that point. The angular size of the image is independent of the mirror's orientation, as long as the observer's position relative to the object (themselves) and the image's size relative to the object remain unchanged.

Consider the vertical extent of the image $H$ . The observer's eye is a point from which the entire image must be visible. The required length of the mirror is determined by the intersection of the cone of rays from the image extremities (top and bottom) to the eye with the mirror plane. For a vertical mirror, the required length is $H/2$ . If the mirror is tilted, the required section of the mirror is still determined by the lines connecting the eye to the top and bottom of the image. The size of this section on the tilted mirror surface might be different from the vertical height, but the minimum vertical extent of the mirror might still be related to $H/2$ . However, the question asks for the minimum height of the mirror, which usually refers to the length of the mirror surface itself.

Let's reconsider the similar triangles. The ratio of the size of the required mirror section to the size of the image is equal to the ratio of the distance from the eye to the mirror to the distance from the eye to the image. Let the distance from the eye to the mirror be $d_m$ (measured perpendicular to the mirror plane). The distance from the eye to the image is $d_i = 2 d_m$ . The size of the image is $H$ . Let the required length of the mirror surface be $L$ . By similar triangles (formed by the eye, the extremities of the image, and the corresponding points on the mirror), the ratio of the mirror length to the image length is $d_m / d_i = d_m / (2 d_m) = 1/2$ . So, the minimum length of the mirror surface required is $L = H/2$ .

This result holds true for a plane mirror regardless of its tilt, as long as the observer is viewing their own image and the mirror is large enough and positioned appropriately. The tilt affects the position and orientation of the required mirror section, but not its minimum length along the mirror surface.

The question asks for the minimum height of the mirror. In the context of a mirror for viewing a full body, this generally refers to the minimum length of the mirror along its surface, which is oriented to reflect the image.

The angle $\theta$ to the vertical does not change the fact that the image is the same size as the object ( $H$ ) and is located behind the mirror. The geometry of similar triangles formed by the eye, the image points, and the mirror surface dictates that the length of the mirror segment needed to view the full image is half the length of the image. Since the image height is $H$ , the minimum length of the mirror is $H/2$ .

The angle of tilt $\theta$ determines the orientation of the mirror and thus the required position of the mirror. For example, if the mirror is tilted, the bottom edge of the mirror might need to be higher or lower than $H_e/2$ vertically, and the top edge might also change its vertical position, but the length of the mirror surface itself required is still $H/2$ .

The phrase "minimum height of mirror needed" most likely refers to the minimum dimension of the mirror surface itself along the direction that captures the full body image.

Final check: The standard derivation for a vertical mirror gives $H/2$ . The similar triangle argument based on the image size and distances holds regardless of the plane mirror's orientation. The length of the mirror surface needed is always half the height of the object being viewed.

Thus, the minimum height (length) of the mirror needed is $H/2$ . The angle $\theta$ is irrelevant to the minimum size requirement for a plane mirror.