- a spherical generator is used to produce a -15.00D surface on glass of refractive index 1.80. The diameter of the cutting tool is 80mm and the radius of the cutting surface is 4 mm. What is the angle between the axis of the tool and the axis of the lens?
To find the angle between the axis of the tool and the axis of the lens, we can first calculate the sagitta of the cutting surface using the given radius and refractive index.
The sagitta is the maximum depth of the cutting surface, and it can be calculated using the formula:
sagitta = (cutting depth) * (1 - (1 / refractive index))
Given:
Radius of cutting surface (r) = 4 mm
Refractive index (n) = 1.80
Substituting the values into the formula:
sagitta = 4 * (1 - (1 / 1.80))
Next, we can calculate the angle using trigonometry. We have a right triangle formed by the sagitta (opposite side), the radius of the cutting tool (adjacent side), and the hypotenuse (radius of the cutting surface). The angle between the hypotenuse and the adjacent side represents the angle between the axis of the tool and the axis of the lens.
Using the tangent function:
tan(angle) = sagitta / radius
Substituting the values:
tan(angle) = (sagitta) / 80
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan((sagitta) / 80)
Now, let's plug in the value of the sagitta calculated in the previous step to find the angle between the axes of the tool and the lens.