There is no relationship between a straight line of sight and the curvature of the Earth. You would need to define that line in relation to the sphere. For example, we can create a straight line through a laser pointer. Thus, emulating a straight line of sight we have at the center of our eyes' focus. I could place it level on the ground, at which point there would be a relationship to define mathematically. I could place it on top of a ladder, or a building, which would change the relationship. I could place the laser at any number of angles, also changing the relationship.
We can create any numbers of mathematic formulae to describe this relationship. So, if we put a laser pointer on the ground, perpendicular to the distance to the center (flat on the ground), we can determine with simple geometry the change in distance of the beam of light to the surface of the Earth, at any particular point. We would be treating the light as it would behave in a vacuum, for the sake of simplicity, as atmospheric refraction can change this.
In this model, we can use a right angle triangle, as we know height squared plus base squared equals the squared length. (HxH + BxB= LxL). Since the Earth is a sphere, we can use its radius at our height (6371 km). That is, there are 6371 kilometers between the surface of the earth at which the laser pointer begins and the center of the Earth at which it is perpendicular. As the Earth is curved, we can expect the beam of light to increase its distance above the surface the more it travels. Let's make it 26.2km, the distance of a marathon run.
So we have a straight line, from the center of the Earth to the surface of 6371km. At this point, we form a right angle with the laser beam extending to a distance of 26.2km. The distance between that point, and the center of the Earth can easily be calculated:
HxH + BxB = LxL
6371x6371 + 26.2x26.2 = LxL
40,590,327.44 = L x L
L = 6371.0538720057924... km
Since the radius of the Earth is 6371km, we see that line of light, or "sight" is 0.0538720057924... km above the Earth. That's about 177 ft. Meaning, if line of sight was 177ft above ground, you could see the ground up to 26.2km away.
That's the math - and the reality is easily observed. When we increase our elevation, we can see further, before being limited by the horizon. If the Earth were flat, we would literally be able to see to the end of the Earth.
The Earth is not flat and anyone can see this. Go 500 ft up above the surface. Then 1000 ft. Then 2000 ft. The higher you go, the further you can see. If the Earth were flat, and there was nothing blocking your view of distant object going higher would actually increase your distance to the object and make it more obscure.
