TE
科技回声
首页24小时热榜最新最佳问答展示工作
GitHubTwitter
首页

科技回声

基于 Next.js 构建的科技新闻平台,提供全球科技新闻和讨论内容。

GitHubTwitter

首页

首页最新最佳问答展示工作

资源链接

HackerNews API原版 HackerNewsNext.js

© 2025 科技回声. 版权所有。

ASK HN: Can any of you solve this?

14 点作者 eam将近 14 年前
I was driving the other day, having lots of time to think, I wonder what percentage of the entire sky was I looking at.

6 条评论

svrocks将近 14 年前
Assumptions: 1) Earth is a sphere with radius r 2) "Sky" is a hollow spherical shell with radius R<p>The surface area of the entire spherical sky is: 4\piR^2<p>This can be represented by a spherical integral that I'm not sure I can write cleanly here.<p>We just need to change the bounds of that integral to find the area of the observable part of that shell. The Intersecting Chord Theorem along with some trigonometry can be used to find these bounds.<p>The answer I get is:<p>(1 - cos(x)) / 2 where x = Arctan(sqrt(r(R-r)) / r)<p>This seems to have the correct asymptotic behavior (as r approaches 0, cos(x) approaches cos(pi/2) = 0, and the answer approaches 50%<p>EDIT: My previous answer assumed the shapes were cones instead of spheres. Sorry about the confusion.
评论 #2574429 未加载
评论 #2574419 未加载
评论 #2574558 未加载
评论 #2574641 未加载
davidhollander将近 14 年前
[edit, Made a mistake, added sanity checks]<p>sin theta = r_earth / (r_earth + altitude)<p>theta = arcsin( r_earth/(r_earth + altitude))<p>p_visible = (2pi - 2*theta)/(2pi)<p>p_visible = 1 - arcsin ( r_earth / (r_earth + altitude))/pi<p>.<p>.<p>Sanity checks:<p>r_earth = 6378.1 X 1000 m<p>altitude = 1m<p>p_visible = .50006<p>.<p>altitude = 10^7 m<p>p_visible = .66766<p>This fits intuitively: the further away you are from the Earth's surface, the more of the sky you can see without the horizon interfering.
mdpm将近 14 年前
most of that is simple geometry (heh). The interesting parts creep in with the 'non-ideal' conditions.<p>You're not just referring to the percentage of the surface area of a sphere (which earth isn't), atmospheric distribution isn't uniform even if we go by volume (or should we be going by density?), then there's the curvature of light in the atmosphere to take into consideration, and the arbitrary descisions as to what height above sea level our observer is standing at, the variable nature of the tropopause ...<p>an interesting problem, but likely more interesting as a mental exercise than in actual execution.
ColinWright将近 14 年前
The question is ill defined - it depends.<p>Define "The Sky" as an enclosing sphere. When it has the same size as the Earth, the percentage of it you can see is 0. As it gets larger, so the percentage you can see goes to 50%.
评论 #2574606 未加载
eam将近 14 年前
I was driving the other day, having lots of time to think, I wondered what percentage of the entire sky was I looking at.
评论 #2574344 未加载
hfinney将近 14 年前
What % of the sky is visible from a point on a sphere.<p>50%, duh.
评论 #2574459 未加载
评论 #2574471 未加载