Reflection Calculator & Graph Data
This calculator determines the properties of a reflected ray given an incident ray and a mirror line. It provides the point of incidence, angles, and the equation of the reflected ray, along with a point on it, which can be used to graph the reflection.
Incident Ray Definition
Define the incident ray by two points it passes through. The ray originates from Point 1 and travels towards Point 2.
Mirror Line Definition
Define the mirror line by two points it passes through.
Understanding Reflection
Reflection is a fundamental phenomenon in optics and physics where a wave (such as light or sound) encounters a surface or boundary that does not absorb the energy of the wave and instead bounces it back. The most common example is the reflection of light from a mirror.
Key Concepts:
- Incident Ray: The ray of light (or other wave) that strikes the surface.
- Reflected Ray: The ray that bounces off the surface after reflection.
- Normal: An imaginary line perpendicular to the surface at the point where the incident ray strikes.
- Point of Incidence: The exact point where the incident ray meets the reflecting surface.
- Angle of Incidence: The angle between the incident ray and the normal.
- Angle of Reflection: The angle between the reflected ray and the normal.
The Law of Reflection:
The Law of Reflection states two crucial points:
- The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
- The angle of incidence is equal to the angle of reflection. (i.e., Angle of Incidence = Angle of Reflection).
This calculator uses these principles to determine the path of the reflected ray.
How This Calculator Works:
You define the incident ray by providing two points it passes through (Point 1 and Point 2). The ray is assumed to originate from Point 1 and travel towards Point 2. You also define the mirror line by two points it passes through. The calculator then performs the following steps:
- Determines the equations of the incident ray and the mirror line.
- Calculates the Point of Incidence (POI): This is the intersection point of the incident ray and the mirror line.
- Calculates the angle of the incident ray and the mirror line relative to the positive x-axis.
- Applies the Law of Reflection: It calculates the angle of the reflected ray based on the incident ray's angle and the mirror's orientation. The angle of incidence (between the incident ray and the normal) is also calculated and shown to be equal to the angle of reflection.
- Derives the equation of the reflected ray and provides an additional point on this ray, which is useful for plotting.
Graphing the Results:
The output of this calculator provides all the necessary data to manually or programmatically graph the incident ray, the mirror, and the reflected ray on a coordinate plane. You can use the provided coordinates and equations to visualize the reflection.
Example Calculation:
Let's use the default values to illustrate:
- Incident Ray: From (1, 5) through (3, 3)
- Mirror Line: From (0, 0) to (5, 0) (a horizontal line along the x-axis)
When you click "Calculate Reflection" with these values, the calculator will determine:
- The incident ray's equation:
y = -x + 6 - The mirror line's equation:
y = 0 - The Point of Incidence (POI):
(6, 0) - The angle of incidence and reflection (which will be 45 degrees in this case).
- The reflected ray's equation:
y = x - 6 - A point on the reflected ray, for example,
(7, 1).
You can then plot these points and lines to see the reflection visually.
Reflection Results
"; resultHTML += "Incident Ray: From (" + x1.toFixed(2) + ", " + y1.toFixed(2) + ") through (" + x2.toFixed(2) + ", " + y2.toFixed(2) + ")"; resultHTML += "Mirror Line: From (" + mx1.toFixed(2) + ", " + my1.toFixed(2) + ") to (" + mx2.toFixed(2) + ", " + my2.toFixed(2) + ")"; resultHTML += "Point of Incidence (POI): (" + x_poi.toFixed(4) + ", " + y_poi.toFixed(4) + ")"; resultHTML += "Angle of Incident Ray (relative to positive x-axis): " + theta_inc_deg.toFixed(2) + " degrees"; resultHTML += "Angle of Mirror Line (relative to positive x-axis): " + theta_mirror_deg.toFixed(2) + " degrees"; resultHTML += "Angle of Incidence (between incident ray and normal): " + angle_incidence_deg.toFixed(2) + " degrees"; resultHTML += "Angle of Reflection (between reflected ray and normal): " + angle_incidence_deg.toFixed(2) + " degrees (equal to angle of incidence)"; resultHTML += "Angle of Reflected Ray (relative to positive x-axis): " + theta_ref_deg.toFixed(2) + " degrees"; if (Math.abs(Math.cos(theta_ref)) < 1e-9) { // Check for near-vertical reflected ray resultHTML += "Equation of Reflected Ray: x = " + x_poi.toFixed(4) + " (Vertical Line)"; } else { resultHTML += "Equation of Reflected Ray: y = " + m_ref.toFixed(4) + "x + " + c_ref.toFixed(4) + ""; } resultHTML += "A Point on the Reflected Ray: (" + x_ref_point.toFixed(4) + ", " + y_ref_point.toFixed(4) + ") (useful for graphing)"; resultHTML += "How to Graph These Results:
"; resultHTML += "- ";
resultHTML += "
- Plot the Incident Ray: Draw a line from (" + x1.toFixed(2) + ", " + y1.toFixed(2) + ") through (" + x2.toFixed(2) + ", " + y2.toFixed(2) + ") extending to the Point of Incidence. "; resultHTML += "
- Plot the Mirror Line: Draw a line passing through (" + mx1.toFixed(2) + ", " + my1.toFixed(2) + ") and (" + mx2.toFixed(2) + ", " + my2.toFixed(2) + "). "; resultHTML += "
- Plot the Point of Incidence: Mark (" + x_poi.toFixed(4) + ", " + y_poi.toFixed(4) + ") on the mirror. "; resultHTML += "
- Plot the Reflected Ray: Draw a line starting from the Point of Incidence (" + x_poi.toFixed(4) + ", " + y_poi.toFixed(4) + ") and passing through the calculated point (" + x_ref_point.toFixed(4) + ", " + y_ref_point.toFixed(4) + "). "; resultHTML += "