Imagine a laser installed in a lighthouse which is perpendicular to earth and the laser is parallel to the earth. Does it touches earth?

 If the laser installed in the lighthouse is parallel to the Earth's surface (assuming Earth is a sphere), the laser would not touch the Earth, except at the point it originates.

This is due to the curvature of the Earth. As the laser extends in a straight line away from the lighthouse, the Earth's surface curves downward relative to the straight path of the laser. Over time, the gap between the laser beam and the Earth's surface increases.

For example:

• Over a distance of 1 kilometer, the Earth's surface would curve away by about 7.8 centimeters (based on Earth's radius of ~6,371 kilometers).
• The farther the laser extends, the larger this gap becomes due to the increasing curvature.

In short, unless the laser points downward (not parallel), it will not touch the Earth's surface after leaving the lighthouse.