218 Figure \ref{fig:product-regions} shows some examples. |
218 Figure \ref{fig:product-regions} shows some examples. |
219 \begin{figure}[t] |
219 \begin{figure}[t] |
220 $$ |
220 $$ |
221 \mathfig{0.5}{triangle/triangle2} |
221 \mathfig{0.5}{triangle/triangle2} |
222 $$ |
222 $$ |
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223 \begin{align*} |
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224 \begin{tikzpicture}[baseline] |
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225 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$}; |
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226 \node (d) at (c.east) [circle through = {(0.25,0)}] {}; |
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227 \foreach \n in {1,2} { |
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228 \node (p\n) at (intersection \n of c and d) {}; |
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229 \fill (p\n) circle (2pt); |
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230 } |
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231 \begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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232 \draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east); |
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233 \end{scope} |
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234 \end{tikzpicture} & = |
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235 \begin{tikzpicture}[baseline] |
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236 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {}; |
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237 \begin{scope} |
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238 \path[clip] (c) circle (1); |
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239 \node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {}; |
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240 \foreach \n in {1,2} { |
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241 \node (p\n) at (intersection \n of c and d) {}; |
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242 } |
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243 \node[left] at (c) {$f$}; |
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244 \path[clip] (d) circle (0.75); |
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245 \foreach \y in {1,0.86,...,-1} { |
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246 \draw[green!50!brown] (0,\y)--(1,\y); |
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247 } |
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248 \end{scope} |
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249 \draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0); |
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250 \end{tikzpicture} \\ |
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251 \begin{tikzpicture}[baseline] |
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252 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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253 \draw[dashed] (c.north) -- (c.south); |
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254 \node[right=6] at (c) {$g$}; |
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255 \node[left=6] at (c) {$f$}; |
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256 \end{tikzpicture} & = |
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257 \begin{tikzpicture}[baseline] |
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258 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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259 \node[right=9] at (c) {$g$}; |
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260 \node[left=9] at (c) {$f$}; |
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261 \draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
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262 \begin{scope} |
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263 \path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north); |
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264 \foreach \y in {1,0.86,...,-1} { |
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265 \draw[green!50!brown] (-1,\y)--(1,\y); |
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266 } |
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267 \end{scope} |
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268 \draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25); |
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269 \end{tikzpicture} \\ |
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270 \begin{tikzpicture}[baseline] |
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271 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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272 \draw[dashed] (c.north) -- (c.south); |
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273 \node[right=18] at (c) {$g$}; |
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274 \node[left=10] at (c) {$f$}; |
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275 \fill (0,0.4) node (p1) {} circle (2pt); |
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276 \fill (0,-0.4) node (p2) {} circle (2pt); |
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277 \begin{scope}[decoration={brace,amplitude=5,aspect=0.5}] |
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278 \draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east); |
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279 \end{scope} |
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280 \end{tikzpicture} & = |
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281 \begin{tikzpicture}[baseline] |
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282 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {}; |
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283 \node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {}; |
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284 \draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south); |
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285 \node[right=18] at (c) {$g$}; |
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286 \node[left=18] at (c) {$f$}; |
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287 \clip (0,0) circle (0.5cm); |
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288 \foreach \y in {1,0.86,...,-1} { |
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289 \draw[green!50!brown] (-1,\y)--(1,\y); |
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290 } |
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291 \end{tikzpicture} |
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292 \end{align*} |
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293 \todo{fourth case} |
223 \caption{Examples of inserting or deleting product regions.} |
294 \caption{Examples of inserting or deleting product regions.} |
224 \label{fig:product-regions} |
295 \label{fig:product-regions} |
225 \end{figure} |
296 \end{figure} |
226 |
297 |
227 |
298 |