Locating Images with Mirror Ray Models

Learning Goals

• Distinguish an object point, an image point, and the path light actually travels.
• Locate plane-mirror images by backward extension of reflected rays.
• Use principal rays to construct images for concave and convex spherical mirrors.
• Apply \(f=R/2\), the mirror equation, and lateral magnification consistently.
• Interpret signs as real or virtual, upright or inverted, enlarged or reduced.
• Predict image changes as an object crosses \(C\), \(F\), or moves toward a mirror.
• Evaluate model limits, experimental evidence, and common ray-diagram errors.

1. What Counts as an Image?

An object point sends light along many paths. A mirror redirects part of that light. An image point is the location where reflected rays actually meet or appear to originate.

• Real image: reflected rays physically pass through the image point. A screen can intercept it.
• Virtual image: reflected rays diverge, but their backward extensions meet. No light travels through the apparent point, so a screen placed there cannot capture it.

Dashed lines behind a mirror are geometric extensions used by an observer's ray model. They are not additional light rays moving through the mirror.

2. Plane Mirrors: Symmetry Across the Mirror Plane

For every object point, reflected rays reaching an observer behave as if they came from a point equally far behind the mirror. Under the sign convention used in this lesson, a real object has \(d_o>0\) and the plane-mirror image has

The image is virtual, upright, and the same size as the object. “Left-right reversal” is better understood as front-back reversal: the image is related to the object by reflection across the mirror plane, not by a physical rotation.

3. Geometry of Spherical Mirrors

A spherical mirror is part of a sphere. Its vertex \(V\) is where the optical axis meets the surface, its center of curvature \(C\) is the sphere's center, and its radius is \(R=VC\). In the paraxial approximation—rays close to the axis and making small angles—the focal point lies halfway between \(V\) and \(C\):

A concave mirror is converging and has \(f>0\). A convex mirror is diverging and has \(f<0\). These signs are part of the model, not optional labels added after a calculation.

6. A Reliable Ray-Diagram Procedure

1. Draw the optical axis, mirror vertex, \(F\), and \(C\) to scale when possible.
2. Draw the object arrow from the axis and begin rays at its tip.
3. Use any two correct principal rays; the third is a consistency check.
4. For a real image, mark where reflected rays actually cross.
5. For a virtual image, extend reflected rays backward with dashed lines.
6. Draw the image from the axis to the intersection and classify location, orientation, and relative size.

The principal rays are convenient representatives of many rays leaving each object point. Blocking one principal ray does not normally erase a corresponding part of the image.

7. Mirror Equation and Magnification

For paraxial rays and a spherical mirror, object distance, image distance, and focal length obey

The signed lateral magnification connects height and distance:

Use the ray diagram to predict the sign and approximate magnitude before calculating. The equation should refine the sketch, not replace physical reasoning.

8. Concave-Mirror Image Regimes

9. What Happens at the Focal Point?

When a real object is at \(d_o=f\), the mirror equation gives \(1/d_i=0\), so the reflected rays are parallel and the ideal image is at infinity. A nearby screen cannot capture a focused image. Moving the object only slightly across \(F\) changes the image from a distant real image to a distant virtual image, so image position is extremely sensitive near the focal point.

This divergence is a model prediction, not an infinite-energy result. Real mirrors have finite aperture, aberrations, and diffraction, and real objects have nonzero depth.

10. Convex Mirrors Always Diverge Reflected Rays

For a real object, a convex mirror has \(f<0\). The reflected rays diverge, and their backward extensions meet behind the mirror between \(V\) and \(F\). Therefore \(d_i<0\), \(m>0\), and \(0

The reduced image allows a wider field of view, which is useful for security and vehicle mirrors. The apparent distance can be misleading because the image is smaller.

11. Projection Test and the Observer

A real image can be formed on a screen because light from each object point converges to a corresponding image point. A virtual image cannot be projected at its apparent location because the rays do not pass there. Yet an eye or camera can still focus the diverging rays and record a virtual image.

Whether an observer sees an image also depends on whether reflected rays enter the pupil. The geometric image location is set by the mirror and object, not by the observer's position, but the visible portion and field of view can change.

12. Mirror Power and Curvature

Optical power is the inverse focal length in meters:

The unit is the diopter, \(\mathrm D=\mathrm{m^{-1}}\). A concave mirror has positive power and a convex mirror has negative power. Greater curvature means smaller \(\lvert R\rvert\), smaller \(\lvert f\rvert\), and larger \(\lvert P\rvert\).

15. Reliable Quantitative Workflow

1. Classify the mirror and predict the image qualitatively.
2. Draw a small ray sketch with \(V\), \(F\), \(C\), and the object.
3. Assign signed \(f\), \(d_o\), and any known \(d_i\) using one convention.
4. Solve the mirror equation symbolically before substituting values.
5. Use \(m=-d_i/d_o=h_i/h_o\) for orientation and size.
6. Translate every sign into words: real or virtual, upright or inverted.
7. Check the result against the ray regime and limiting behavior.